How can one show that the polynomial:
$$ f= (2+i)X^4 + 5X^2 - 3i \hspace{15px}\text{with coefficients in } \mathbb{Z}[i] $$
is reducible/irreducible. I know about the Eisenstein criterion and the reduction criterion, but I can't seem to get a grasp on how to apply these in a useful way.
Multiplying with $(2-i)/5$ we obtain a monic polynomial of degree $4$. Setting $x=X^2$ it is a quadratic polynomial. Suppose it is of the form $(x+a+bi)(x+c+di)$ and compare coefficients. One obtains easy Diophantine equations in $a,b,c,d$. One of them is $$ 5(a^2+2a-b^2+b)=3, $$ a contradiction, because $5$ does not divide $3$. Hence the polynomial is indeed irreducible.