Let $R$ be a commutative ring with a multiplicative id. The question reads:
Let $r$ be an element of a ring $R$. Show that, in the polynomial ring $R[X]$, the polynomial $1+rX$ is a unit if and only if $r$ is nilpotent. Is it possible for the polynomial $1 + X$ to be a product of two non-units?
I've done the first part, it wasn't too bad. But I'm quite stuck with the second, and any hints or pointers would be appreciated. Am I looking for a proof or counterexample? What kind of thing should I be looking at? Etc.