Solvability of a quintic polynomial.

Question

I have the following polynomial that I would like to obtain an analytical solution to

$$
f(x)=3x(-2x^5+7x^4-7x^3+2x+1)
$$

I am not certain how to approach this, and even to test if a solution exists. I have not studied Galois theory although I see that this is relevant. It is of order 6 in $x$, but can it be reduced to a quintic if I extract $x$ in this way?

Answer 1

pari/gp calculation

? polgalois(-2*x^5 + 7*x^4 - 7*x^3 + 2*x + 1)
%2 = [120, -1, 1, "S5"]

shows that this polynomial has $S_5$ Galois group. Galois theory says that a polynomial with that Galois group cannot be solved using radicals.