8x^3 - 27 = 0
8x^3 = (2x)^3
(2x)^3 = 3^3
2x = 3 --> 2x - 3 is a factor
Divide by 2x - 3
(8x^3 - 27)/(2x - 3) = 4x^2 + 6x + 9
Now solve for 0 and factor
4x^2 + 6x + 9 = 0
x = [-6 +/- sqrt(-108)]/8
x = (3/4)(-1 +/- i√3]
So the solutions are:
x = 3/2
x = -(3/4)(1 + i√3)
x = -(3/4)(1 - i√3