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GCSE Maths · Algebra

Expanding brackets GCSE Questions and Worked Answers

Expanding brackets rewrites a product as a sum of terms. In 3(x + 2), there are three copies of both x and 2, so the result is 3x + 6.

Foundation & Higher6 worked examples11 original questions
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Start with the meaning

What you need to know about expanding brackets

Imagine three bags. Each contains an unknown number of counters, which we call x, plus two extra counters. One bag contains x + 2 counters. Three bags contain 3(x + 2): writing a number beside a bracket means multiply. Counting the two kinds of counters separately gives 3x + 6. Expanding changes how the same total is written, not its value.

See the idea first

Every part is repeated

A term is one part of a sum, such as x or 2. Both terms belong to each bag.

Three bags(x + 2) + (x + 2) + (x + 2)write every copy
Collect each kindx + x + x + 2 + 2 + 2three unknown amounts and six extras
Same total3x + 6the expanded expression
From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Collect like terms before using brackets

What the problem asks: Simplify 4x + 3y − x + 2y.

How to solve it: Like terms have exactly the same letter part. Four x amounts minus one x leaves 3x; three y amounts plus two y gives 5y. The result is 3x + 5y. Do not combine x with y, or x with x².

Multiply one bracket

What the problem asks: Expand 5(x + 4).

How to solve it: Multiply both terms by 5: 5x + 20. This is five copies of the whole bracket.

Subtract a whole bracket

What the problem asks: Expand −2(x − 3).

How to solve it: Multiply each signed term by −2: −2x + 6. Multiplying two negatives gives a positive.

Multiply two brackets

What the problem asks: Expand (x + 2)(x + 5).

How to solve it: Multiply x and 2 each by the entire second bracket, giving x² + 5x + 2x + 10. Combine 5x and 2x to get x² + 7x + 10.

Expand and simplify

What the problem asks: Simplify 3(x + 2) + 2x.

How to solve it: First get 3x + 6 + 2x. Like terms contain the same letter part, so 3x + 2x becomes 5x; the answer is 5x + 6.

A reliable routine

For a factor multiplying a sum

Distribution means multiplying the factor by every term inside the bracket. It works because the whole sum is being multiplied. With two brackets, distribute each term of the first across every term of the second.

  1. Identify every term, keeping its sign attached.
  2. Multiply every required pair: number parts together and letter parts together.
  3. Write all products before collecting like terms.
  4. Check that no term was missed. Substitute a simple value as a useful error check, not a proof.

Check: x × x = x², whereas x + x = 2x. Expanding an expression does not require finding x: there is no equation to solve.

Fully worked

Expanding brackets GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

One bracket

2 marks
Question

Expand 4(2x + 3).

Multiply 4 by both terms.

4(2x+3)=8x+124(2x+3)=8x+12

For x = 1, both forms give 20, a useful check.

Example 2

Negative multiplier

2 marks
Question

Expand −3(2x − 5).

Keep the negative signs in the products.

−3(2x−5)=(−3)(2x)+(−3)(−5)-3(2x-5)=(-3)(2x)+(-3)(-5) =−6x+15=-6x+15

The second product is positive.

Example 3

A letter outside

2 marks
Question

Expand 2x(3x+4)2x(3x+4).

The outside factor is the whole 2x.

2x(3x+4)=6x2+8x2x(3x+4)=6x^2+8x

The first product contains x × x; the second contains only one x.

Example 4

Two brackets

3 marks
Question

Expand and simplify (x+3)(x−4)(x+3)(x-4).

Multiply x, then 3, by both terms.

(x+3)(x−4)=x2−4x+3x−12(x+3)(x-4)=x^2-4x+3x-12 =x2−x−12=x^2-x-12

The middle terms have the same letter part. Each table cell below multiplies its row label by its column label, so no product is missed.

Four products from (x + 3)(x − 4)
×x−4
xx²−4x
+33x−12
Example 5

A squared bracket

3 marks
Question

Expand (x+5)2(x+5)^2.

A square means two copies multiplied, not just squaring each term.

(x+5)2=(x+5)(x+5)(x+5)^2=(x+5)(x+5) =x2+5x+5x+25=x^2+5x+5x+25 =x2+10x+25=x^2+10x+25

The two cross-products account for 10x.

Example 6

Three brackets

Higher only4 marks
Question

Expand (x+1)(x+2)(x−1)(x+1)(x+2)(x-1).

Multiply two brackets first; pairing x + 1 and x − 1 is convenient.

(x+1)(x−1)=x2−x+x−1(x+1)(x-1)=x^2-x+x-1 =x2−1=x^2-1

The two x terms cancel. Then multiply the result by the remaining bracket.

(x2−1)(x+2)=x3+2x2−x−2(x^2-1)(x+2)=x^3+2x^2-x-2

Keep the remaining factor until the final multiplication.

11 original questions · total 25 marks

Expanding brackets GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 30 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
1

Collect signed like terms

3 marks

Simplify 7a − 2b − 3a + 5b.

Show worked answer
7a−3a=4a7a-3a=4a −2b+5b=3b-2b+5b=3b

So the simplified expression is 4a + 3b. Reorder terms with their signs attached.

2

Equal groups

1 mark

Expand 6(x + 2).

Show worked answer
6(x+2)=6x+126(x+2)=6x+12

Both parts are repeated six times.

3

Subtract inside

1 mark

Expand 3(4x − 7).

Show worked answer
3(4x−7)=12x−213(4x-7)=12x-21

Multiply the negative term too.

4

Minus one

2 marks

Expand −(5x − 8).

Show worked answer

The outside multiplier is −1.

−(5x−8)=−5x+8-(5x-8)=-5x+8

Both signs change.

5

A letter factor

2 marks

Expand 4y(2y + 3).

Show worked answer
4y(2y+3)=8y2+12y4y(2y+3)=8y^2+12y

Two y factors make y².

6

Collect afterwards

2 marks

Expand and simplify 2(x + 7) + 3(x − 1).

Show worked answer
2(x+7)+3(x−1)=2x+14+3x−32(x+7)+3(x-1)=2x+14+3x-3 =5x+11=5x+11

Collect the x terms and numbers separately.

7

Two positive brackets

2 marks

Expand (x + 4)(x + 6).

Show worked answer
(x+4)(x+6)=x2+6x+4x+24(x+4)(x+6)=x^2+6x+4x+24 =x2+10x+24=x^2+10x+24

Include all four products.

8

Two negative terms

3 marks

Expand (x − 2)(x − 7).

Show worked answer
(x−2)(x−7)=x2−7x−2x+14(x-2)(x-7)=x^2-7x-2x+14 =x2−9x+14=x^2-9x+14

The last product is positive.

9

Different first terms

3 marks

Expand (2x + 3)(x − 5).

Show worked answer
(2x+3)(x−5)=2x2−10x+3x−15(2x+3)(x-5)=2x^2-10x+3x-15 =2x2−7x−15=2x^2-7x-15

Combine only the two x terms.

10

Explain an error

2 marks

A student says (x − 4)² = x² + 16. Correct the expansion.

Show worked answer
(x−4)2=(x−4)(x−4)(x-4)^2=(x-4)(x-4) =x2−4x−4x+16=x^2-4x-4x+16 =x2−8x+16=x^2-8x+16

The student missed both cross-products.

11

Three factors

Higher only4 marks

Expand (x − 1)(x + 1)(x + 3).

Show worked answer
(x−1)(x+1)=x2+x−x−1(x-1)(x+1)=x^2+x-x-1 =x2−1=x^2-1

The two x terms cancel.

(x2−1)(x+3)=x3+3x2−x−3(x^2-1)(x+3)=x^3+3x^2-x-3

Multiply the first result by every term in the final bracket.

Examiner-style feedback

Common expanding brackets mistakes

Multiplying only the first term

A factor outside applies to the whole bracket, including constants and negative terms.

Combining unlike terms

x² and x describe different powers. 3x² + 2x cannot be collected into 5x².

Squaring each part separately

(a + b)² includes two ab products. Write two brackets to see them.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

  1. A bracket is one grouped amount.
  2. Multiply every required pair.
  3. Keep signs with terms.
  4. Collect only matching letter parts.
Quick answers

Expanding brackets FAQ

What is the difference between expanding and factorising?

Expanding writes a product as a sum. Factorising reverses that process, expressing a sum as a product.

Is expanding two brackets Higher only?

No. Products of two binomials are assessed across tiers. Products of three binomials are Higher content.

Build connected skills

What to revise next

Browse all GCSE Maths study guidesReview the worked examplesTry the questions again
Content standards

Curriculum and rights review

A4: expanding single and double brackets across tiers, with a labelled Higher extension to three binomials. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references