Irreducible in $(\mathbb{Z}[i])[x]$ Prove that $p(x)=x^3-6x^2+4ix+1+3i$ is irreducible in $(\mathbb{Z}[i])[x]$.
I'm not so clear on irreducibility in specific instances like this one. I know I need to show that if $p(x)=q(x)r(x)$ then W.L.O.G. $q(x)$ is a unit. Since $\mathbb{Z}[i]$ is an integral domain, it follows that $(\mathbb{Z}[i])[x]$ is also an integral domain, so the units are simply $\{1,-1,i,-i\}$. Some back and forth feedback would be preferred to a straightaway proof.
I know that $\deg(p(x))=3$ $\Rightarrow \deg(q(x))+\deg(r(x))=3$. The question becomes how to show $\deg(q(x))=0$.

Hint $ $ If $\,p = \pi\bar \pi$ divides all non-extremal coefs of $\,f(x)\,$ and $\,p\,||\,N(f(0))$ then the Eisenstein irreducibility test applies for prime $\,\pi\,$ or $\,\bar \pi,\,$ since e.g. $\,\pi^2\!\mid f(0)\,\Rightarrow\,p^2\! =N(\pi^2)\mid N(f(0)).$
