What is expanding brackets?
Expanding brackets means multiplying everything inside a bracket by the term (or terms) outside. It is the reverse process of factorising and is one of the most frequently tested skills across both Foundation and Higher GCSE papers.
There are four main types you need to master:
- Single brackets — multiply the term outside by every term inside
- Double brackets (FOIL) — multiply every term in the first bracket by every term in the second
- Perfect squares — expanding (ax + b)²
- Difference of two squares — (a + b)(a − b) = a² − b²
Single brackets
Multiply the term outside the bracket by every term inside. Pay close attention to negative signs.
Worked Example 1
Expand: −3x(2x − 5)
1
Multiply −3x by 2x: −6x²
2
Multiply −3x by −5: +15x (negative × negative = positive)
Answer−6x² + 15x
Double brackets — FOIL
Use FOIL: First, Outer, Inner, Last. Multiply every term in the first bracket by every term in the second, then collect like terms.
Worked Example 2
Expand and simplify: (x + 5)(x − 3)
1
First: x × x = x²
2
Outer: x × −3 = −3x
3
Inner: 5 × x = +5x
4
Last: 5 × −3 = −15
5
Collect like terms: x² − 3x + 5x − 15 = x² + 2x − 15
Answerx² + 2x − 15
Worked Example 3 — Perfect square
Expand and simplify: (3x − 1)²
1
Write as two brackets: (3x − 1)(3x − 1)
2
FOIL: 9x² − 3x − 3x + 1 = 9x² − 6x + 1
Answer9x² − 6x + 1
Worked Example 4 — Difference of two squares
Expand: (2x + 7)(2x − 7)
1
Recognise the pattern (a + b)(a − b) = a² − b²
2
a = 2x, b = 7: (2x)² − 7² = 4x² − 49
Answer4x² − 49
Practice questions
Work through each question before checking the answers below.
Foundation (Grade 3–5)
Q1Expand: 3(2x − 5)Foundation
Q2Expand: −4x(x + 3)Foundation
Q3Expand and simplify: (x + 5)(x − 2)Foundation
Q4Expand and simplify: (3x − 1)(2x + 4)Foundation
Q5Expand: (x + 3)²Foundation
Higher (Grade 5–7)
Q6Expand and simplify: (2x − 5)(2x + 5)Higher
Q7Expand and simplify: (x + 4)(x − 4)Higher
Q8Expand and simplify: (3x + 2)²Higher
Q9Expand and simplify: (x + 3)(x − 3)(x + 1)Higher
Q10Show that (n + 1)² − (n − 1)² = 4n for all integer values of nHigher
Higher — Hard (Grade 8–9)
Q11Expand and simplify: (2x + 1)³Grade 8–9
Q12Expand and simplify: (x + y)² − (x − y)²Grade 8–9
Q13Show that (2n + 3)² − (2n − 1)² = 8(2n + 1)Grade 8–9
Q14Expand and simplify: (x² + 3)(x² − 3)Grade 8–9
Q15Expand and simplify: (3x − 2)² − (x + 1)(x − 1)Grade 8–9
Answers
Foundation (Q1–Q5)
Q16x − 15
Q2−4x² − 12x
Q3x² + 3x − 10(FOIL: x² − 2x + 5x − 10)
Q46x² + 10x − 4(FOIL: 6x² + 12x − 2x − 4)
Q5x² + 6x + 9
Higher (Q6–Q10)
Q64x² − 25(difference of two squares)
Q7x² − 16(difference of two squares)
Q89x² + 12x + 4((3x+2)(3x+2) = 9x²+6x+6x+4)
Q9x³ + x² − 9x − 9(first expand (x+3)(x−3) = x²−9, then ×(x+1))
Q10(n+1)²−(n−1)² = n²+2n+1−(n²−2n+1) = 4n
Higher — Hard (Q11–Q15)
Q118x³ + 12x² + 6x + 1((2x+1)²=4x²+4x+1; ×(2x+1) = 8x³+12x²+6x+1)
Q124xy((x+y)²=x²+2xy+y²; (x−y)²=x²−2xy+y²; difference = 4xy)
Q13(2n+3)²−(2n−1)² = 4n²+12n+9−(4n²−4n+1) = 16n+8 = 8(2n+1)
Q14x⁴ − 9(difference of two squares: (x²)²−3²)
Q158x² − 12x + 5(9x²−12x+4 − (x²−1) = 8x²−12x+5)
Common mistakes
Common Mistake 1
Only multiplying the first term in a single bracket
Students expand 3(2x − 5) as 6x − 5, forgetting to multiply the −5. Every term inside must be multiplied by the term outside: 3(2x − 5) = 6x − 15.
Common Mistake 2
Sign errors with negatives outside a bracket
Expanding −2(x − 4): students often write −2x − 8. This is wrong. −2 × −4 = +8, so the answer is −2x + 8. Always apply the sign of the outer term to every inner term.
Common Mistake 3
Missing the middle terms in a perfect square
Students write (x + 3)² = x² + 9. This is wrong. Always expand fully: (x + 3)(x + 3) = x² + 3x + 3x + 9 = x² + 6x + 9. The middle term is never zero unless the signs differ.
Common Mistake 4
Not collecting like terms after FOIL
After expanding (x + 4)(x − 2) = x² − 2x + 4x − 8, students leave the answer as four separate terms instead of simplifying to x² + 2x − 8. Always check for like terms and combine them.
Common Mistake 5
Confusing difference of two squares with a perfect square
(x + 4)(x − 4) = x² − 16, not x² + 0x − 16 written messily, and definitely not (x − 4)². The DOTS pattern gives a positive squared term, a negative squared term, and no middle term.
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