Foundation

What “Expanding Brackets” Actually Means

Core Concept

When you expand double brackets like (x+2)(x+3), you’re multiplying every term in the first bracket by every term in the second bracket. Nothing disappears. Nothing gets skipped. The goal is to remove the brackets and write the expression as a simplified polynomial — usually in the form ax² + bx + c.

Think of it this way. The brackets are a shorthand. (x+2)(x+3) is just saying: take the quantity (x+2) and multiply it by the quantity (x+3). When you expand, you’re doing that multiplication in full and writing out every product.

There are two reliable methods: FOIL and the grid method. Both give the same answer. FOIL is faster once you know it. The grid method is more visual and harder to mess up when you’re learning. This guide covers both so you can pick whichever clicks.

4 individual multiplications needed to expand any double bracket
3 terms in the final simplified answer for most double brackets
2 methods that always work: FOIL and the grid method
1 most common mistake: forgetting to multiply the inner and outer terms
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Why This Topic Comes Up in Exams

Expanding double brackets is a foundational algebra skill. It’s tested directly in GCSE maths, A-Level, and most high school and college algebra courses. It also underpins factorising quadratics — you can’t reliably go backwards (factorise) if you don’t understand the expansion process. Getting this right now pays dividends across the entire algebra curriculum.

Method 1

The FOIL Method — What Each Letter Means

FOIL is an acronym. Each letter tells you which pair of terms to multiply next. Work through all four in order and you won’t miss anything.

F First x · x = x²

Multiply the first terms in each bracket

O Outer x · 3 = 3x

Multiply the outer terms — first of bracket 1, last of bracket 2

I Inner 2 · x = 2x

Multiply the inner terms — last of bracket 1, first of bracket 2

L Last 2 · 3 = 6

Multiply the last terms in each bracket

That’s the full set. Four multiplications. Then you write the results side by side and simplify by collecting like terms. That’s the whole process.

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Why FOIL Works

FOIL is just a structured application of the distributive law. Every term in the first bracket must be multiplied by every term in the second bracket. With two terms in each bracket, that’s 2 × 2 = 4 products. FOIL gives you a fixed order to do them in — so you don’t accidentally skip one. Miss the inner or outer pair and your answer will be wrong every single time.

Worked Example

(x+2)(x+3) — Worked Through Step by Step

Here’s the full expansion, step by step, using FOIL. Each line shows exactly what’s being multiplied and why.

1

Write out the expression

Start with what you’ve been given. Nothing changes yet — just set it up clearly.

2

Multiply the FIRST terms: x × x

The first term in bracket 1 is x. The first term in bracket 2 is x. x × x = x². Write that down.

3

Multiply the OUTER terms: x × 3

The outer terms are the first term of bracket 1 and the last term of bracket 2. x × 3 = 3x. Add it to your running total.

4

Multiply the INNER terms: 2 × x

The inner terms are the last term of bracket 1 and the first term of bracket 2. 2 × x = 2x. Write it down next.

5

Multiply the LAST terms: 2 × 3

The last term of bracket 1 is 2. The last term of bracket 2 is 3. 2 × 3 = 6. That’s all four multiplications done.

6

Collect like terms and simplify

You now have: x² + 3x + 2x + 6. The middle two terms are both “x terms” — they can be added together. 3x + 2x = 5x. Final answer: x² + 5x + 6.

Start: (x + 2)(x + 3)
F: x · x = x²
O: x · 3 = 3x
I: 2 · x = 2x
L: 2 · 3 = 6
Expanded: x² + 3x + 2x + 6
Simplified: x² + 5x + 6
3x + 2x = 5x (like terms collected)

The answer to (x+2)(x+3) is x² + 5x + 6. That’s it. Four multiplications, one simplification step.

— The complete process, nothing more needed
Method 2

The Grid Method — More Visual, Just as Reliable

Some students find FOIL hard to track mentally. The grid method makes the same four multiplications visible in a table — so it’s much harder to miss one. Set it up like this:

Put the terms of the first bracket down the left column and the terms of the second bracket along the top row. Then fill in each cell by multiplying the row header by the column header.

x +3
x x² 3x
+2 2x 6

Read out the four cells: x², 3x, 2x, 6. Write them together: x² + 3x + 2x + 6. Collect like terms: x² + 5x + 6. Same answer as FOIL — always will be.

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Which Method Should You Use?

  • FOIL is faster once it’s automatic. Good for timed exams when you’re confident.
  • Grid method is more systematic. Better when you’re just learning, or when brackets have more than two terms each.
  • Both are accepted in exam mark schemes. Use whichever gives you the right answer reliably — that’s the one to stick with.
Simplifying

Collecting Like Terms — The Step Students Rush and Get Wrong

After you expand, you usually have four terms. Two of them will be “like terms” — terms with the same variable and power. You combine them by adding their coefficients. The x² and the constant stay as they are.

After Expansion Like Terms to Combine Simplified Result
x² + 3x + 2x + 6 3x + 2x = 5x x² + 5x + 6
x² + 5x + 4x + 20 5x + 4x = 9x x² + 9x + 20
x² + 2x − 3x − 6 2x − 3x = −x x² − x − 6
x² − 4x + 4x − 16 −4x + 4x = 0 x² − 16

That last row is a special case — the difference of two squares. When both middle terms cancel out, you’re left with just two terms. It’s worth recognising that pattern. It comes up a lot.

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Watch the Signs When Collecting

If either bracket contains a subtraction — like (x − 3) — the signs carry through. A negative times a positive gives a negative. A negative times a negative gives a positive. Track the sign of every term you write down, not just the number. That’s where most errors happen: the multiplication is right, but the sign gets dropped.

Practice

More Examples — Same Method, Different Numbers

The process is always the same: four multiplications, collect like terms. Here are worked examples with different bracket types so you can see the pattern across a range of questions.

Example: (x + 4)(x + 5)

Worked

F: x × x = x²
O: x × 5 = 5x
I: 4 × x = 4x
L: 4 × 5 = 20

Expanded: x² + 5x + 4x + 20
Simplified: x² + 9x + 20

Example: (x + 3)(x − 2)

Worked — with subtraction

F: x × x = x²
O: x × (−2) = −2x
I: 3 × x = 3x
L: 3 × (−2) = −6

Expanded: x² − 2x + 3x − 6
Simplified: x² + x − 6

Note: −2x + 3x = +x. The signs matter — don’t drop them.

Example: (2x + 1)(x + 3)

Worked — coefficient on x

F: 2x × x = 2x²
O: 2x × 3 = 6x
I: 1 × x = x
L: 1 × 3 = 3

Expanded: 2x² + 6x + x + 3
Simplified: 2x² + 7x + 3

When the coefficient on x isn’t 1, the x² term picks up that coefficient too. 2x × x = 2x², not x².

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For More Practice Problems

The Khan Academy multiplying binomials exercise set gives you unlimited practice problems with immediate feedback — all free. Work through at least ten before your next exam and the method will be automatic.

Common Errors

Mistakes That Cost Marks — and How to Avoid Them

Expansion Errors

  • Only multiplying the first terms and ignoring the rest — writing x² + 6 instead of expanding all four products
  • Forgetting the inner and outer terms entirely — the most common single error
  • Dropping a negative sign when one bracket contains a subtraction
  • Writing 2x² when the first bracket is (x+2) — confusing the constant with a coefficient
  • Not multiplying every term in bracket 1 by every term in bracket 2

Simplification Errors

  • Not combining the like x-terms — leaving the answer as x² + 3x + 2x + 6 instead of simplifying
  • Adding the x² and x terms together — x² and 5x are not like terms and cannot be combined
  • Getting the sign wrong when collecting: −2x + 3x = +x, not −x
  • Forgetting to write the constant term in the final answer
  • Mixing up the order: writing 5x + x² instead of x² + 5x + 6 (doesn’t lose marks, but looks untidy)
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The Single Best Check: Substitute a Number

After you expand, substitute a simple number — like x = 1 or x = 2 — into both the original bracketed expression and your expanded answer. If they give the same value, your expansion is correct. For example: (1+2)(1+3) = 3 × 4 = 12. Expanded: 1² + 5(1) + 6 = 1 + 5 + 6 = 12. They match. If they don’t match, you’ve made an error somewhere in the expansion.

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Common Questions

Expanding Brackets — Questions Answered

What is the answer to (x+2)(x+3)?
The expanded and simplified answer is x² + 5x + 6. You get there by multiplying every term in the first bracket by every term in the second bracket: x×x = x², x×3 = 3x, 2×x = 2x, 2×3 = 6. That gives x² + 3x + 2x + 6. Then collect the like x-terms: 3x + 2x = 5x. Final answer: x² + 5x + 6. But if you need to submit working in an assignment or exam, the method is what earns you marks — not just the answer.
What is the FOIL method and when should I use it?
FOIL stands for First, Outer, Inner, Last. It’s a memory aid for expanding two brackets, each with two terms. F: multiply the first terms. O: multiply the outer terms. I: multiply the inner terms. L: multiply the last terms. Use it whenever you’re expanding two binomials (two-term brackets). It doesn’t work directly for brackets with three or more terms — for those, the grid method or full distribution is more reliable.
What’s the difference between expanding and factorising?
Expanding and factorising are inverse operations. Expanding takes brackets and removes them: (x+2)(x+3) → x² + 5x + 6. Factorising goes the other way — it takes a quadratic expression and rewrites it as a product of two brackets: x² + 5x + 6 → (x+2)(x+3). If you understand how expansion works, factorising is much more approachable because you already know what the answer is supposed to look like.
Why do I get four terms when expanding but only three in the final answer?
Because two of the four terms are like terms — they contain the same variable to the same power (both are x-terms in this case). You combine them by adding their coefficients. 3x and 2x are both x-terms, so 3x + 2x = 5x. The x² term and the constant term (6) have no like terms to combine with, so they stay as they are. That’s why you start with four terms but end with three.
What happens when one of the brackets has a negative sign — like (x−3)?
The process is identical, but you need to carry the negative sign through every multiplication involving that term. For example, expanding (x+2)(x−3): F: x×x = x². O: x×(−3) = −3x. I: 2×x = 2x. L: 2×(−3) = −6. Expanded: x² − 3x + 2x − 6. Collect like terms: −3x + 2x = −x. Final answer: x² − x − 6. The sign handling is where most errors happen — write each term with its sign explicitly before collecting.
How do I check my expansion is correct?
Substitute a number into both the original expression and your expanded answer. They should give the same result. Using x = 2 for (x+2)(x+3): original gives (2+2)(2+3) = 4×5 = 20. Expanded answer x²+5x+6 gives 4+10+6 = 20. They match. This works for any expansion. Use x = 1 or x = 2 — small numbers keep the arithmetic manageable. If the values don’t match, go back and check each multiplication in turn.
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Yes. Our maths tutors and academic writers can help with bracket expansion, factorising quadratics, solving equations, and any other algebra topic — from GCSE level through to university. We also support maths tutoring, calculus homework help, statistics assignments, and physics and geometry. Every session or assignment is tailored to your level and the specific question you’re working on.
Summary

Expanding Brackets Is Four Multiplications and One Simplification

That’s the whole thing. (x+2)(x+3) gives you x² + 3x + 2x + 6. Collect the like terms. You get x² + 5x + 6. Every double bracket question follows the same process — the numbers change, the method doesn’t.

The two things that catch students out are: missing one of the four multiplications (usually the inner or outer pair) and getting signs wrong when brackets contain subtraction. Both are fixable with the substitution check: plug in a number, verify both expressions agree.

If you need support with maths homework, exam preparation, or algebra coursework at any level, Smart Academic Writing offers algebra and maths homework help, online tutoring, and high school assignment support — all tailored to your specific topic and level.