Posted question rewritten

Question

lim
x-> -1

108(x^2+2x)(x+1)^3
____________________

(x^3+1)^3 (x-1)

Answer 1

look at the bottom, this factors to

[(x+1)(x^2 - x + 1)]^3

or (x+1)^3(x^2 - x + 1)^3

so you have
lim(108(x^2 + 2x)/[(x^2 - x + 1)^3(x-1) as x ---> -1
= 108(1-2)/[(1 + 1 + 1)^3(-2)]
= -108/-54
= 2