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

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A fraction (4x^3-32)/[x^3+(x+2)^3] is not defined whenever the denominator is equal to 0.

x^3+(x+2)^3 = 0

=> x^3 + x^3 + 12x + 6x^2 + 8 = 0

=> 2x^3 + 6x^2 + 12x + 8 = 0

=> x^3 + 3x^2 + 6x + 4 = 0

=> x^3 + x^2 + 2x^2 + 6x + 4 = 0

=> x^3 + x^2 + 2x^2 + 4x + 2x + 4 = 0

=> x^2(x +1) +2x(x + 1) + 4(x +1) =0

=> (x^2 + 2x + 4)(x + 1) = 0

x1 = -1

x2 = [-2 + sqrt (4 - 16)] / 2

=> x2 = -1 + i*sqrt 12 / 2

=> x2 = -1 + i*sqrt 3

x3 = -1 - i*sqrt 3

We see that for the values x2 and x3, the fraction takes on the form 0/0.

The values of x for which the fraction is not defined are -1 , -1 + i*sqrt 3 and -1 - i*sqrt 3

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