Proving Irreducibility of $x^4-16x^3+20x^2+12$ in $\mathbb Q[x]$

Note that any factorization will be of the form: $$x^4 -16x^3+20x^2+12 = (x^2+2ax+2b)(x^2+2cx+2d)$$ Here $a,b,c,d$ are integers. We have used Gauss' lemma to note that if $f(x)$ is reducible over $\Bbb Q$ then it is reducible over $\Bbb Z$. The factors of two come from the fact that after reducing mod $2$, our factorization must turn into a factorization of $x^4$.

Expanding our expression gives us the following equations to be satisfied. $$\begin{align*} 2a + 2c &= -16\\ 2b+2d + 4ac &= 20\\ 4bc+4ad &= 0\\ 4bd &= 12 \end{align*}$$ The last of these implies that the pair $\{b,d\}$ is either $\{1,3\}$ or $\{-1, -3\}$. From this setup it is elementary to check that no choice of integers will work