Polynomial $x^5 + 5x^2 +1$ is irreducible or not

Question

Which way I can determine whether the polynomial $x^5 + 5x^2 +1$ is irreducible over $\mathbb Q$ or not?

Mod $p$ Irreducibility Test and Eisenstein's criterion not applicable here.

Which way I should proceed?

Answer 1

Try $x=y-1$. It should help by the Eisenstein's criterion.

Answer 2

$7^5+5\cdot 7^2+1 = 17053$ which is a prime number. Thus the polynomial is irreducible by Cohn's irreducibility criterion.