Expanding Double Brackets

Exam code: 4024

Written by: Ashika|Reviewed by: Caroline Carroll|Updated 2 July 2026

Expanding Two Bracketsvideo

Expanding Two Brackets

Expanding two brackets

How do I expand two brackets?

Every term in the first bracket must be multiplied by every term in the second bracket
- Expanding (x + 1)(x + 3) requires 4 multiplications in total
  - x(x + 1) + 3(x + 1)
  - x×x + 1x + 3x + 3
  - x² + 4x + 3

Using a grid

You can use a grid for a visual way to expand brackets
To expand (x + 1)(x + 3), write out the brackets as row and column headings of a grid
- They can be in either direction
- Remember to write the appropriate sign in front of each term
x
+1
x

+3
For each cell in the grid, multiply the term in the row heading by the term in the column heading
x
+1
x
x²
x
+3
3x
3
Add together all the terms inside the grid to get the answer
- x² + x + 3x + 3
Collect like terms
x² + 4x + 3

Using FOIL

Every term in the first bracket must be multiplied by every term in the second bracket
- Expanding (x + 1)(x + 3) requires 4 multiplications in total
A good way to remember all the multiplications is FOIL
- F = First: multiply together the first terms in each bracket
- O = Outside: multiply the first term in the first bracket by the last term in the last bracket
  - Visually, these are the outer terms
- I = Inside: multiply the last term in the first bracket by the first term in the last bracket
  - Visually, these are the inner terms
- L = Last: multiply together the last terms in each bracket
It helps to put negative terms in brackets when multiplying
Simplify the final answer by collecting like terms (if there are any)

How do I expand when there are multiple variables?

All the same rules and methods apply as when there is just one variable
Remember to only simplify like terms
For example: $(3 x + 2 y) (4 x - 6 y)$
- Expanding: $12 x^{2} - 18 x y + 8 x y - 12 y^{2}$
- The $x y$ terms can be combined
- $12 x^{2} - 10 x y - 12 y^{2}$

Worked example

(a) Expand $(2 x - 3) (x + 4)$ .

Answer:

Using FOIL, multiply together the first, outer, inner and last terms

$F O I L 2 x \times x + 2 x \times 4 + (- 3) \times x + (- 3) \times 4$

Simplify each term

$2 x^{2} + 8 x - 3 x - 12$

Collect like terms (the 8x and -3x)

$2 x^{2} + 5 x - 12$

(b) Expand $(x - 3) (3 x - 5)$ .

Answer:

Using FOIL, multiply together the first, outer, inner and last terms

$F O I L x \times 3 x + x \times (- 5) + (- 3) \times 3 x + (- 3) \times (- 5)$

Simplify each term

$3 x^{2} - 5 x - 9 x + 15$

Collect like terms (the -5x and -9x)

$3 x^{2} - 14 x + 15$

Worked example

Expand $(3 r + 2 t) (5 t - 8 r)$ .

Answer:

Expand using your chosen method, here we will use a grid

	$3 r$	$+ 2 t$
$5 t$
$- 8 r$

Work out the term in each place in the grid by multiplying

	$3 r$	$+ 2 t$
$5 t$	$15 r t$	$10 t^{2}$
$- 8 r$	$- 24 r^{2}$	$- 16 r t$

So the expanded expression is

$10 t^{2} + 15 r t - 16 r t - 24 r^{2}$

The $r t$ terms can be combined

$10 t^{2} - r t - 24 r^{2}$

Expanding Squared Bracketsvideo

Expanding Squared Brackets

Expanding squared brackets

How do I expand a bracket squared?

Remember that a square number is a number multiplied by itself
Write (x + 3)² as (x + 3)(x + 3) and use one of the methods above
- With FOIL: (x + 3)(x + 3) = x²+ 3x + 3x + 9
- Then collect like terms: x² + 6x + 9

Worked example

Expand ${(2 x + 3)}^{2}$ .

Answer:

Remember that the answer is not (2x)² + 3²
Rewrite the expression as two separate brackets multiplied together

$(2 x + 3) (2 x + 3)$

Using FOIL, multiply together the first, outer, inner and last terms

$F O I L 2 x \times 2 x + 2 x \times 3 + 3 \times 2 x + 3 \times 3$

Simplify each term

$4 x^{2} + 6 x + 6 x + 9$

Collect like terms (the 6x and 6x)

$4 x^{2} + 12 x + 9$

	x	+1
x	x²	x
+3	3x	3

Expanding Two Bracketsvideo

Expanding two brackets

How do I expand two brackets?

Using a grid

Using FOIL

How do I expand when there are multiple variables?

Expanding Squared Bracketsvideo

Expanding squared brackets

How do I expand a bracket squared?

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