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Expanding and Factorising Brackets (Factoring) Explained

Multiply out single and double brackets, take out common factors and use the difference of two squares. GCSE factorising and US factoring.

  • ⏱ 3 min read
  • ✏️ 3 worked examples
  • 🎯 5 practice questions
  • 📶 Beginner
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Curriculum fit: GCSE Maths Algebra (Foundation and Higher). US: Algebra I polynomial arithmetic and factoring.

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What Is It?

Expanding means multiplying out brackets. Factorising is the reverse: writing an expression as a product, usually by taking out a common factor or by writing a quadratic as two brackets.

These skills are essential for solving equations and simplifying algebra.

x²3x2x6x+ 3x+ 2(x + 3)(x + 2)= x² + 2x + 3x + 6= x² + 5x + 6
Draw a rectangle with sides (x + 3) and (x + 2). The four smaller areas are the four terms you get by multiplying out.

Key Formulas and Ideas

  • Single bracket: a(b + c) = ab + ac.
  • Double brackets: (x + p)(x + q) = x² + (p + q)x + pq.
  • Common factor: take out the highest common factor of all the terms.
  • Difference of two squares: x² − a² = (x − a)(x + a).

Worked Examples

Example 1

Expand a Double Bracket

Question

Expand (x − 3)(x + 5).

  1. x × x = x², x × 5 = 5x, −3 × x = −3x, −3 × 5 = −15.
  2. Collect like terms: 5x − 3x = 2x.

✓ Answer x² + 2x − 15

Example 2

Common Factor

Question

Factorise 6x + 9.

  1. The highest common factor of 6 and 9 is 3.
  2. Take 3 out: 3(2x + 3).

✓ Answer 3(2x + 3)

Example 3

Difference of Two Squares

Question

Factorise x² − 25.

  1. x² − 25 is x² − 5².
  2. = (x − 5)(x + 5).

✓ Answer (x − 5)(x + 5)

Common Mistakes to Avoid

  • Multiplying only the first term in the bracket.
  • Forgetting the middle terms when expanding double brackets.
  • Not taking out the highest common factor, giving a partly factorised answer.
  • Sign errors with negative terms.

Practice Questions

0 of 5 answered
Q1Expand 5(2x − 3).
Q2Expand (x + 3)(x + 5).
Q3Factorise 9x + 12.
Q4Factorise x² − 25.
Q5Expand (x − 2)².

Frequently Asked Questions

How can I check a factorisation?

Expand your answer. If you get back the original expression, it is correct.

What should I factorise first?

Look for a common factor in every term first. Then check whether what is left is a quadratic you can factorise further, or a difference of two squares.

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Sources and Review

Reviewed by StudyMorrow editorial review on 5 Nov 2026.