Here is a question from an old exam:
1.
a) Let $R$ be a UFD and let $A=a_{0}x^{m}+...+a_{m}\ne 0 , B=b_{0}x^{n}+...+b_{n} \ne 0$ in $R[x]$ with $\gcd(a_{0},...,a_{m})\in R^{*}$, $\gcd(b_{0},...,b_{n})\in R^{*}$. For $C=AB=c_{0}x^{m+n}+...+c_{m+n}$. Show that $\gcd(c_{0},...,c_{m+n})\in R^{*}$.
b) With $K=\mathbf{C}(t)$, show that $A=x^{2011}+x+t$ in $K[x]$ is irreducible.
I didn't know how to begin, so I asked an older student and he told me to use $f:R\rightarrow S=R/pR$. In retrospect I still don't know how to begin despite the hint. (For b he told me to use $R=\mathbf{C[t]}$ together with a).). Help is greatly appreciated.