Multiply out single and double brackets, simplify the result, and reverse the process by taking out a common factor. It is on Foundation and Higher, aimed at grade 4. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Expanding is multiplying out brackets: 3(x+4)=3x+12, and (x+2)(x+5)=x2+7x+10 by four products (FOIL / grid / smiley face — pick a layout and use it every time). Factorising is the reverse: take out the highest common factor, or for a quadratic find two numbers that multiply to the last term and add to the middle coefficient.
Always expand your factors in the margin to check. x2+5x+6=(x+2)(x+3). Difference of two squares: x2−9=(x−3)(x+3). An expression that is already a product of two brackets is factorised; an expression with + or − between terms is not done yet.
Expanding is multiplying every term in the first bracket by every term in the second (FOIL / grid). Factorising is the reverse: take out the HCF of a linear expression, or find two numbers with the right product and sum for a quadratic.
Objectives
3(x + 2) = 3 × x + 3 × 2 = 3x + 6.
A single bracket is this topic. Two brackets, such as (x + 2)(x + 3), come with quadratics.
3x + 6 = 3(x + 2).
Key points
Expand a single bracket by multiplying every term inside by the term outside.
Expand two single brackets and collect like terms, watching negative signs.
Expand double brackets such as (x+7)(x−3) using a grid or the four-multiplication method.
Factorise by taking out the highest common factor of numbers and letters.
Worked examples
Example 1
Expand 3(x + 1).
2 marks
Multiply 3 by x to get 3x.
Multiply 3 by 1 to get 3.
3(x + 1) = 3x + 3.
Answer: 3x+3
Common slips
Slip: Multiply 5 by x to get 3x. That changes a number from the question, so it answers a different question.
Slip: Multiply 6 by 1 to get 3. That changes a number in the working, so the step no longer follows from the one before.
Slip: 7(x + 1) = 3x + 3. That does not match the working above it.
Example 2
Factorise 6x + 6.
2 marks
The common factor of 6x and 6 is 6.
6x ÷ 6 = x and 6 ÷ 6 = 1.
6x + 6 = 6(x + 1).
Answer: 6(x+1)
Common slips
Slip: The common factor of 8x and 6 is 6. That changes a number from the question, so it answers a different question.
Slip: 9x ÷ 6 = x and 6 ÷ 6 = 1. That changes a number in the working, so the step no longer follows from the one before.
Slip: 10x + 6 = 6(x + 1). That does not match the working above it.
Example 3
Expand 5(x − 5).
2 marks
Multiply 5 by x to get 5x.
Multiply 5 by −5 to get −25.
5(x − 5) = 5x − 25.
Answer: 5x-25
Common slips
Slip: Multiply 7 by x to get 5x. That changes a number from the question, so it answers a different question.
Slip: Multiply 8 by −5 to get −25. That changes a number in the working, so the step no longer follows from the one before.
Slip: 9(x − 5) = 5x − 25. That does not match the working above it.
Questions and answers
GCSE Maths expanding & factorising questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Factorise 4x − 20.
Show answer
Answer: 4(x-5)
The common factor of 4x and 20 is 4.
4x ÷ 4 = x and −20 ÷ 4 = −5.
M1 — The common factor of 4x and 20 is 4.
A1 — Answer 4(x-5)
Question 2 Easy2 marks
Expand 3(x + 12).
Show answer
Answer: 3x+36
Multiply 3 by 12 to get 36.
3(x + 12) = 3x + 36.
M1 — Multiply 3 by 12 to get 36.
A1 — Answer 3x+36
Question 3 Medium2 marks
Factorise 15x + 6.
Show answer
Answer: 3(5x+2)
The common factor of 15x and 6 is 3.
15x ÷ 3 = 5x and 6 ÷ 3 = 2.
M1 — The common factor of 15x and 6 is 3.
A1 — Answer 3(5x+2)
Question 4 Medium2 marks
Expand 5(x + 7).
Show answer
Answer: 5x+35
Multiply 5 by 7 to get 35.
5(x + 7) = 5x + 35.
M1 — Multiply 5 by 7 to get 35.
A1 — Answer 5x+35
Question 5 Hard2 marks
Factorise 5x − 55.
Show answer
Answer: 5(x-11)
The common factor of 5x and 55 is 5.
5x ÷ 5 = x and −55 ÷ 5 = −11.
M1 — The common factor of 5x and 55 is 5.
A1 — Answer 5(x-11)
Question 6 Hard2 marks
Expand 5(x + 10).
Show answer
Answer: 5x+50
Multiply 5 by 10 to get 50.
5(x + 10) = 5x + 50.
M1 — Multiply 5 by 10 to get 50.
A1 — Answer 5x+50
Question 7 Exam-style2 marks
Factorise 9x + 15.
Show answer
Answer: 3(3x+5)
The common factor of 9x and 15 is 3.
9x ÷ 3 = 3x and 15 ÷ 3 = 5.
9x + 15 = 3(3x + 5).
M1 — The common factor of 9x and 15 is 3.
A1 — Answer 3(3x+5)
Common mistakes
Most lost marks on this topic are one slip.
Multiplying only the first term in the bracket.
The outside factor multiplies every term.
Taking out a factor that does not divide every term.
The factor has to be common.
Dropping the sign when you factorise 3x − 6.
The second term is −2 inside the bracket, not +2.
Spot the slip
The common factor of 3x and 6 is
3
x
6
Show the slip
3 3 divides both terms. x does not divide 6.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Write out all four products for double brackets before collecting — a correct set of four earns a method mark even if you add wrongly.
Check a factorised answer by expanding it in your head; it must give back exactly what you started with.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.