Problem: Suppose that $K$ is a field and that $f$ and $g$ are relatively prime in $K[x]$. Show that $f- yg$ is irreducible in $K(y)[x]$.
My attempt: Consider the polynomial $f-gY\in (K[x])[Y]$. This polynomial is irreducible. Indeed, if it were not, since its degree is over $(K[x])[Y]$ is $1$, $f-yg=(a-by)c$ where $a, b, c\in K[x]$ and the degree of $c$ over $K[x]$ is at least $1$. Then $f-ac=(g-bc)Y$, therefore $f=ac$ and $g=bc$ which contradicts $f$ and $g$ relatively prime.
Now, because $f-gY$ irreducible, if $y$ transcendental over $K$, $f-gy$ will be transcendental over $V[x]$, where $V$ is the k-vectorial space generated by $\lbrace y^n; n\in \mathbb{Z} \rbrace \cup \lbrace 1\rbrace$.
Finally, by Gauss's lema $f-gy$ is irreducible over $K(y)[x]$, since $K(y)$ is the fraction field of $V$.
I don't know if my proof is correct or not... moreover, I don't know how I can prove the statament when $y$ is algebraic.
Comments: This exercise is 5.4 of the book a course in Galois theory by Garling.