Solve the equation $x^3-6x-6=0$ 
Evaluate the roots of
$$x^3-6x-6=0$$

I solved it using Cardano's method, but I'm looking for other elementary approaches through substitutions and properties of polynomials.
Thanks.

Hint:
$x^3-6x-6=x^3-3bcx+b^3+c^3$
$$bc=-2, b^3+c^3=-6$$
$$b=-\sqrt[3]{2}, c=-\sqrt[3]{4}$$
So $(b+c)^3=b^3+c^3+3bc(b+c)$, then number $b+c$ - solution of $x^3-3bcx+b^3+c^3=0$
