What is x for the fraction (4x^3-32)/[x^3+(x+2)^3] if not defined ?

A fraction is undefined if and only if it's denominator is
cancelling.

So, we'll have to find out the roots of the
denominator.

We notice that we can write the numerator as a
difference of cubes, after factorizing by 4:

4(x^3 - 8) =
4(x - 2)(x^2 + 2x + 4)

We'll write the denominator as a sum
of cubes:

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

We'll expand the square and we'll remove the
brackets inside the second factor's brackets:

x^2 - x(x+2)
+ (x+2)^2 = x^2 - x^2 - 2x + x^2 + 4x + 4

We'll combine and
eliminate like terms:

x^2 - x(x+2) + (x+2)^2 = (x^2 + 2x +
4)

We'll re-write the
fraction:

4(x - 2)(x^2 + 2x + 4)/(2x+2)(x^2 + 2x +
4)

We'll simplify:

4(x -
2)/2(x+1) = 2(x-2)/(x+1)

The fraction is undefined for x +
1 = 0

x = -1

The
value of x for the fraction if undefined is x =
-1.