I want to prove that the polynomial
$$Y^3+XY^2+X^3Y+X$$
is irreducible in $\mathbb{C}[X,Y]$.  

If I try to write it as product of two polynomials $g$ and $h$, I can see that in order for $Y^3$ to appear, $g$ must contain a term $\alpha Y^2$ and $h$ a term $1/\alpha Y$ or vice versa and similarly one must contain $\beta X$ and the other $1/\beta$ etc., but the cases seem too complicated.  

Also, because the coefficient field is $\mathbb{C}$, I don't see how usual tricks like Eisenstein's can work.