Let $h(x)=x^4+12x^3+14x^2-12x+1$, and let $p>5$ be a prime.
I want to show $h(x)$ factors into 2 quadratics $\mod p$, if $p \equiv 9,11 \mod 20$, while $h(x)$ factors mod $p$ into 4 linear factors, if $p \equiv 1,19 \mod 20$. I can show $h(x)$ is irreducible if $p \equiv 3,7 \mod 10$.