How to solve the equation $x^2+2=4\sqrt{x^3+1}$? From the Leningrad Mathematical Olympiad, 1975:

Solve $x^2+2=4\sqrt{x^3+1}$.

In answer sheet only written $x=4+2\sqrt{3}\pm \sqrt{34+20\sqrt{3}}$. 
How to solve this?

HINT.-The given answer is root of the quadratic equation
$$x^2-2(4+2\sqrt3)x+c=0$$ where $c$ is certain constant. It follows 
$$x=4+2\sqrt3\pm\sqrt{{(4+2\sqrt3)^2-c}}=4+2\sqrt3\pm\sqrt{34+20\sqrt3}$$
You get so the value of $c$ from which you can verify the solution.
