Consider the change of variable
[tex]x^3=t[/tex]
The equation becomes
[tex]t^2+t+1=0[/tex]
This equation has solutions
[tex]t=\dfrac{-1\pm\sqrt{1-4}}{2}=\dfrac{-1\pm\sqrt{-3}}{2}[/tex]
Now we should find the corresponding x values by reverting the variable change:
[tex]x^3=t \implies x = \sqrt[3]{t}[/tex]
Using complex numbers, every number has three cubic roots, so the solutions are
[tex]x_{1,2,3}=\sqrt[3]{\dfrac{-1+\sqrt{-3}}{2}},\quad x_{4,5,6}=\sqrt[3]{\dfrac{-1-\sqrt{-3}}{2}}[/tex]
So, all six roots are complex.
