Solving a six-degree polynomial of the form $ax^6+bx^3+g$.

Question

I read that it is not always possible to solve but from Wikipedia:

Some sixth degree equations, such as $ax^6 + dx^3 + g = 0$, can be solved by factorizing into radicals, but other sextics cannot.

So how do you solve this? I've stumbled upon this equation where; $a = n^2$, $d=m^2n-2p^2$, and $g=m^4$. Where the $a,d,g$ corresponds to the above mentioned sextic-equation and $m,n,p$ are input variables. I apologize for that ton of variables in use, might cause some confusion. Is this possible using Galois Theory or by any other means?

Answer 1

Hint: Make the substitution $y=x^3$. Then $ay^2+dy+g=0$. This equation an be solved by radicals and then the solutions of the original equation are obtained by taking 3-rd roots of unity.