Well, those are incorrect.
(x + 1) (x² - x + 1) = x³ + 1, not (x+1)³ .

(x + 1)³ = x³ + 3x² + 3x + 1



Factor the x³+1 in the denominator.

x³+1 = (x+1)(x² - x + 1)

Since the x³+1 in the denominator is cubed, that should give a factor in the denominator which cancels with (x+1)³ in the numerator.