Is $f(X)=(2+i)X^4+5X^2-3i$ irreducible over $\mathbb{Z}[i]$?

Question

Is $f(X)=(2+i)X^4+5X^2-3i$ irreducible over $\mathbb{Z}[i]$?

I would be pleased if you could tell me how to start.

Thanks.

Answer 1

Hint $ $ Apply Eisenstein to the reverse ("reciprocal") polynomial $\, x^4 f(x^{-1})\,$ using $\,p = 2+i$.

Note that the reversal map $\, f\mapsto x^d f(x^{-1}),\ d=\deg f\,$ is multiplicative, being the product of two multiplicative maps, namely $\,f(x)\mapsto f(x^{-1}),\,$ and $\,f(x)\mapsto x^{\,\deg f}\ $ (an "exponential" of the additive degree map). Being multiplicative, it preserves (ir)reducibility.