Expanding Double Brackets –

Expanding double brackets is a fundamental algebra skill tested on both Foundation and Higher tier GCSE Maths papers. You will need it for factorising quadratics, solving equations, and graph work, so it is worth practising until the method is automatic.

What Is Expanding Double Brackets?

When you expand double brackets you multiply two linear expressions together to produce a quadratic expression. For example, (x + 3)(x + 5) becomes x² + 8x + 15. The process uses the distributive law twice: every term in the first bracket must be multiplied by every term in the second bracket, giving four partial products that are then collected and simplified.

There are two popular methods. The FOIL method labels the four multiplications as First, Outside, Inside, Last. The grid method sets up a 2 x 2 table with one bracket across the top and the other down the side, then fills in each cell. Both give the same answer, so use whichever you find clearer.

A special case worth memorising is the difference of two squares: when the two brackets have the same terms but opposite signs, the middle terms cancel and you are left with a² - b².

Key Formulas

Step-by-Step Method

1. Multiply the first terms of each bracket together to get the x² term.
2. Multiply the outer pair of terms (first term of the first bracket by the last term of the second bracket).
3. Multiply the inner pair of terms (last term of the first bracket by the first term of the second bracket).
4. Multiply the last terms of each bracket together to get the constant term.
5. Collect the two middle terms (outer + inner) and write the simplified expression.

Worked Example 1 — Foundation Level

Question: Expand and simplify (x + 4)(x + 6).

Working:

Step 1 — First: x times x = x².

Step 2 — Outside: x times 6 = 6x.

Step 3 — Inside: 4 times x = 4x.

Step 4 — Last: 4 times 6 = 24.

Step 5 — Collect like terms: x² + 6x + 4x + 24 = x² + 10x + 24.

Answer: x² + 10x + 24

Worked Example 2 — Higher Level

Question: Expand and simplify (3x - 2)(2x + 5).

Working:

Step 1 — First: 3x times 2x = 6x².

Step 2 — Outside: 3x times 5 = 15x.

Step 3 — Inside: (-2) times 2x = -4x.

Step 4 — Last: (-2) times 5 = -10.

Step 5 — Collect like terms: 6x² + 15x - 4x - 10 = 6x² + 11x - 10.

Answer: 6x² + 11x - 10

Worked Example 3 — Exam Style

Question: Show that (x + 7)(x - 7) - (x + 3)² = -58 - 6x. (4 marks)

Working:

Expand (x + 7)(x - 7) using the difference of two squares: x² - 49.

Expand (x + 3)²: (x + 3)(x + 3) = x² + 3x + 3x + 9 = x² + 6x + 9.

Subtract: (x² - 49) - (x² + 6x + 9) = x² - 49 - x² - 6x - 9 = -6x - 58.

This is the same as -58 - 6x. QED.

Answer: Shown: both sides simplify to -6x - 58.

Common Mistakes

• Forgetting the inner or outer product. Students sometimes write only three terms instead of four. Using FOIL labels or a grid prevents this.
• Sign errors with negatives. In (x - 3)(x + 5), the inner product is -3 times x = -3x, not +3x. Track signs carefully at every multiplication.
• Writing (x + 4)² as x² + 16. You must expand as (x + 4)(x + 4) = x² + 8x + 16. The middle term 2ab is always present when the bracket is squared.

Exam Tips

• If the question says "expand and simplify," you must collect like terms or you will lose a mark.
• Use the grid method if you keep making errors with FOIL — it lays out all four products visually.
• Check your expansion by substituting x = 1 into both the brackets and the expanded form; the two results should match.

Practice Questions

Q1 (Foundation): Expand and simplify (x + 5)(x + 2).

Q2 (Foundation): Expand and simplify (x - 3)(x + 8).

Q3 (Higher): Expand and simplify (4x + 1)(2x - 3).

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Related Topics

• Expanding Brackets
• Factorising Expressions
• Simplifying Expressions

Summary

• Expanding double brackets means multiplying every term in the first bracket by every term in the second.
• Use FOIL (First, Outside, Inside, Last) or a 2 x 2 grid to organise the four products.
• Always collect like terms after expanding.
• The difference of two squares is a shortcut: (a + b)(a - b) = a² - b².
• Check your answer by substituting a simple value such as x = 1 into both forms.