Question

Factorise the following:

p(a² + a – b) + q(b – a – a²)

Answer 1

Given expression: p(a² + a – b) + q(b – a – a²)

Step-wise calculation:

1. Rearrange the terms inside the parentheses for the second part:

q(b – a – a²) = q(–a² – a + b)

2. So the expression becomes:

p(a² + a – b) + q(–a² – a + b)

3. Group terms by (a²), (a) and (b):

= pa² + pa – pb – qa² – qa + qb

4. Combine like terms:

= (pa² – qa²) + (pa – qa) + (–pb + qb) 

= (p – q)a² + (p – q)a + (q – p)b

5. Factor out (p – q) where possible:

= (p – q)(a² + a) + (q – p)b

6. Note that (q – p)b = –(p – q)b, so:

= (p – q)(a² + a) – (p – q)b

7. Factor out (p – q) completely:

= (p – q)(a² + a – b)