Solve for x:
x^3+3 x^2+3 x+3 = 0
Subtract 2 from both sides:
x^3+3 x^2+3 x+1 = -2
Factor x^3+3 x^2+3 x+1 into a perfect cube:
(x+1)^3 = -2
Taking cube roots gives (-2)^(1/3) times the third roots of unity:
x+1 = (-2)^(1/3) or x+1 = -2^(1/3) or x+1 = -((-1)^(2/3) 2^(1/3))
Subtract 1 from both sides:
x = (-2)^(1/3)-1 or x+1 = -2^(1/3) or x+1 = -((-1)^(2/3) 2^(1/3))
Subtract 1 from both sides:
x = (-2)^(1/3)-1 or x = -1-2^(1/3) or x+1 = -((-1)^(2/3) 2^(1/3))
Subtract 1 from both sides:
Answer: |x = (-2)^(1/3)-1 or x = -1-2^(1/3) or x = -1-(-1)^(2/3) 2^(1/3)