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