Question

Factorise the following:

(x² + 2x)² – 18(x² + 2x) + 45

Answer 1

Given expression: (x² + 2x)² – 18(x² + 2x) + 45

Step-wise calculation:

1. Let (p = x² + 2x), substituting gives: p² – 18p + 45

2. Factorize the quadratic in (p):

Find two numbers whose product is 45 and sum is 18.

Those numbers are 15 and 3.

p² – 18p + 45 = p² – 15p – 3p + 45 

p² – 18p + 45 = p(p – 15) – 3(p – 15) 

p² – 18p + 45 = (p – 15)(p – 3)

3. Substitute back (p = x² + 2x): 

(x² + 2x – 15)(x² + 2x – 3)

4. Factorise each quadratic:

For (x² + 2x – 15), factors of –15 adding to 2 are 5 and –3: (x + 5)(x – 3)

For (x² + 2x – 3), factors of –3 adding to 2 are 3 and –1: (x + 3)(x – 1)