In order to factorize the given expression, we find to find two numbers p and q such that:

p + q = 12, pq = -45

Clearly,

15 – 3 = 12, 15 (-3) = -45

Therefore, split 12x as 15x – 3x

Therefore,

x² + 12x – 45 = x² + 15x – 3x – 45

= x (x + 15) – 3 (x + 15)

= (x – 3) (x + 15)