Factoring by grouping

Statement

Factor: (a) $3a(a^2 - 5) + 7(a^2 - 5)$; (b) $4(3d + 2) - 7d(3d + 2)$.

Solution

In both cases a common bracketed factor appears, which we can pull out.

(a) The common factor is $(a^2 - 5)$: $3a(a^2 - 5) + 7(a^2 - 5) = (a^2 - 5)(3a + 7).$

(b) The common factor is $(3d + 2)$: $4(3d + 2) - 7d(3d + 2) = (3d + 2)(4 - 7d).$

$\boxed{(a^2 - 5)(3a + 7)\ ;\quad (3d + 2)(4 - 7d)}$