Value of an expression with cube root radical What is the value of the following expression?
$$\sqrt[3]{\ 17\sqrt{5}+38}  - \sqrt[3]{17\sqrt{5}-38}$$

The expression strangely reminds of Cardano's formula for the cubic equation
$$x=\sqrt[3]{-\frac q2-\sqrt{\frac{q^2}4+\frac{p^3}{27}}}+\sqrt[3]{-\frac q2+\sqrt{\frac{q^2}4+\frac{p^3}{27}}}.$$
Identifying, we have 
$$q=-76,p=3,$$
and $x$ is a root of $$x^3+3x-76=0.$$
By inspection (trying values close to $\sqrt[3]{76}$), $\color{green}4$ is a solution.
As $$x^3+3x-76=(x-4)(x^2+4x+19),$$ the other roots are complex.
