Irreducibility of the following Polynomial over $\mathbb{Q}$

Question

Take the polynomial $x^4 + 10x^2 + 1$. Is this irreducible over $\mathbb{Q}$? If so, what is the best way to show this? I know it can be rewritten: $$ (x^2 + 5)^2 -24 $$ Which can be simplified over the irrationals, but not over $\mathbb{Q}$. How could I use this to show irreducibility? (if it has anything to do with it)

Answer 1

Hint: After checking there are no rational roots, you can look for factors of the form $x^2 + a x \pm 1$. Note also that if $p(x)$ is a factor, so is $p(-x)$.