I want to show that $$f=x^4+2y^2x^2+yx+5y$$ is irreducible over $\mathbb C[x,y]$.
My answer is, $f$ is in $\mathbb C[x,y]=\mathbb C[y][x]$, and let $R=\mathbb C[y]$, then $f$ is irreducible over $\mathbb C[y]$ (because if we let $c=y$, then $2y^2$, $y$, $5y$ is divisible by $c$, but $1$ is not divisible by $c$, and $5y$ is not divisible by $c^2$).
I don't know how to do next, but in my guess, $\mathbb C$ is $\text{UFD},$ so $\mathbb C[y]=R$ is also $\text{UFD}.$ And $f$ in $R[x]$ is primitive(Since $f$ is monic), and so on $\dots$
Is there any proposition about relation between this primitive, and irreducible?
Thank you!