Since w is a complex cube root of unity, w satisfies z^3 = 1
So w^3 = 1
w^3 - 1 = 0
(w -1)(w^2 + w + 1) = 0
But w is complex, so w^2 + w + 1 = 0
Using this for that question,
(1 + w)^3(1 + 2w + 2w^2)
= (-w^2)^3(1 + w + w^2 + w + w^2)
= -w^6(1 - 1 - 1)
= -1(-1)
= 1
Basically for these...