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

p + q = -4, pq = -21

Clearly,

3 – 7 = -4, 3 (-7) = -21

Therefore, split (-4x) as 3x – 7x

Therefore,

x² + 4x – 21 = x² + 3x – 7x – 21

= x (x + 3) – 7 (x + 3)

= (x – 7) (x + 3)