Two (Double) bracket expansion

What is double bracket expansion?

When you need to expand two brackets multiplied together, such as (x + 3)(x + 5), every term in the first bracket must multiply every term in the second bracket. This gives you four separate multiplications, which you then simplify by collecting like terms.

There are four popular methods taught in schools. All produce the same answer — choose the one that clicks for you.

The core rule: Every term in the first bracket × every term in the second bracket. Four products total. Then collect like terms.

The four methods

Method 1: FOIL

FOIL is an acronym that names the four multiplications in order: First, Outer, Inner and Last. It works only for two-term × two-term brackets.

Expand (x + 3)(x + 5)

First: x × x = x²

Outer: x × 5 = 5x

Inner: 3 × x = 3x

Last: 3 × 5 = 15

= x² + 5x + 3x + 15 (combine)

= x² + 8x + 15 (collect like terms)

The two middle terms (5x and 3x) are like terms — add them to get 8x

Method 2: The grid (box) method

Split each bracket into its terms and build a multiplication table. Every cell is one product. This is the most reliable method for keeping track of all four multiplications and is especially useful when coefficients are involved.

Expand (2x + 3)(x − 4)

= 2x² + 3x − 8x − 12

= 2x² − 5x − 12

Add all four cells: 2x² + 3x + (−8x) + (−12). Collect the x terms: 3x − 8x = −5x.

Method 3: Arc (smiley face) method

Draw four curved arcs connecting each pair of terms to be multiplied. The two outer terms form one arc, the two inner terms another, and the two "diagonal" pairs complete the four products.

Expand: (x + 4)(x − 2)

= x² − 2x + 4x − 8

= x² + 2x − 8

Method 4: Long multiplication layout

Write the expansion like a standard long multiplication sum, distributing one bracket fully across the other in two rows before adding the columns. This method maps exactly to how you'd multiply two-digit numbers and is very reliable for keeping signs organised.

Expand (3x − 2)(2x + 5)

Step 1: 3x × (2x + 5) = 6x² + 15x

Step 2: −2 × (2x + 5) = −4x − 10

Add: 6x² + 15x − 4x − 10

= 6x² + 11x − 10

Multiply the whole second bracket by each term in the first bracket, one row at a time, then add the rows.

Worked examples

Example 1 : Both terms positive

Expand (x + 2)(x + 7)

= x² + 7x + 2x + 14

= x² + 9x + 14

Example 2: One negative term

Expand (x + 6)(x − 3)

= x² − 3x + 6x − 18

= x² + 3x − 18

Example 3: Both terms negative

Expand (x − 4)(x − 5)

= x² − 5x − 4x + 20

= x² − 9x + 20

Example 4: Coefficients involved

Expand (2x + 3)(3x − 4)

= 6x² + 9x − 8x − 12

= 6x² + x − 12

Example 5: Difference of two squares

Expand (x + 5)(x − 5)

= x² − 5x + 5x − 25

= x² − 25

This is the difference of two squares identity: (a + b)(a − b) = a² − b². The two middle terms always cancel. Spot this pattern — it comes up constantly in factorisation.