National 5MathsSQARevision NotesAlgebraic SkillsExpansion of BracketsDouble Brackets

Double Brackets (SQA National 5 Maths): Revision Note

Exam code: X847 75

Written by: Roger B

Reviewed by: Dan Finlay

Updated on 1 December 2025

Expanding double brackets

How do I expand double brackets?

By double brackets we mean an expression such as $(x + 1) (x + 3)$
- Every term in the first bracket must be multiplied by every term in the second bracket
You can do this by changing the double brackets into a pair of single brackets
- Then use the usual method for expanding single brackets
Split the terms in the first bracket, and multiply the second bracket by each of them separately
- $(x + 1) (x + 3) = x (x + 3) + 1 (x + 3)$
Then expand and simplify the single brackets
- $= x^{2} + 3 x + x + 3$
- $= x^{2} + 4 x + 3$

Examiner Tips and Tricks

This method will work for expanding any pair of double brackets.

But many students prefer one of the methods below (using 'FOIL' or a grid).

How do I expand double brackets using FOIL?

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 double brackets using a grid?

You may prefer a more visual method using a grid
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

	x	+1
x	x²	x
+3	3x	3

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 and simplify $(2 x - 3) (x + 4)$ .

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

Answer:

Part (a)

We will do this using FOIL

Multiply together the first, outer, inner and last terms

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

Carry out the multiplications

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

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

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

Part (b)

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

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

The $r t$ terms can be combined

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

Worked Example

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

Answer:

The double bracket can be split to give two single brackets

Don't forget the minus sign in front of the 3 in the first bracket

$(x - 3) (3 x - 5) - 5 x (x + 2) = x (3 x - 5) - 3 (3 x - 5) - 5 x (x + 2)$

Expand the single brackets in the usual way

Be careful with the minus signs

$= 3 x^{2} - 5 x - 9 x + 15 - 5 x^{2} - 10 x$

Collect like terms (the two x² terms and the three x terms)

$- 2 x^{2} - 24 x + 15$

Expanding squared brackets

How do I expand a bracket squared?

Remember that a square number is a number multiplied by itself
To expand (x + 3)²
- Write it as (x + 3)(x + 3)
- Then use one of the methods above
  - E.g. with FOIL: (x + 3)(x + 3) = x²+ 3x + 3x + 9
  - Then collect like terms: x² + 6x + 9

Examiner Tips and Tricks

Do not make the common mistake of saying (x + 3)² is x² + 3².

To see that this cannot be true, try substituting in x = 1 :

You would get (1 + 3)² = 4² = 16 on the left
but 1² + 3² = 1 + 9 = 10 on the right

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

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

Expand the double brackets, e.g. by using FOIL

Multiply together the first, outer, inner and last terms

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

Carry out the multiplications

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

Collect like terms (the 6x and 6x)

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

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Double Brackets (SQA National 5 Maths): Revision Note

Expanding double brackets

How do I expand double brackets?

Examiner Tips and Tricks

How do I expand double brackets using FOIL?

How do I expand double brackets using a grid?

How do I expand when there are multiple variables?

Worked Example

Worked Example

Expanding squared brackets

How do I expand a bracket squared?

Examiner Tips and Tricks

Worked Example

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Dan Finlay