Question

Factorise the following:

12(x + y)² – x – y – 1

Answer 1

Given expression: 12(x + y)² – x – y – 1

Step-wise calculation:

1. Let p = x + y.

Rewrite the expression in terms of p: 12p² – p – 1.

2. Factor the quadratic expression 12p² – p – 1.

To factor: Find two numbers that multiply to 12 × (–1) = –12 and add to –1.

These numbers are –4 and 3 because –4 × 3 = –12 and –4 + 3 = –1.

3. Rewrite the middle term: 12p² – 4p + 3p – 1.

4. Factor by grouping: 4p(3p – 1) + 1(3p – 1).

5. Factor out (3p – 1): (4p + 1)(3p – 1).

6. Replace p back with (x + y): (4(x + y) + 1)(3(x + y) – 1).