How to prove that $\mathbb R [x]$ is a UFD I could not believe that my professor asked this on the exam especially since this is intro-level abstract algebra. My idea was to somehow find a norm function and use the inequality $N(r) < N(d)$ in $m = qd + r$. I erased that and said: "since we know that $\mathbb{R}$ is an UFD we can show that $\mathbb{R[x]}$ is an UFD." We only had 30 minutes to do per problem and this was one of the unusually hard problems. 

You may prove that $\mathcal{N}(f) = \deg(f)$ is a Euclidean norm on $\mathbb{R}[x]$ (more generally $k[x]$ for any field $k$), thus $\mathbb{R}[x]$ is a Euclidean domain $\Rightarrow$ PID $\Rightarrow$ UFD.
And the logic you wrote down is correct: A field is (trivially) a UFD, since all elements are units, and it is usually proved in classes that $R[x]$ is a UFD if $R$ is a UFD, so your teacher should have given you points for that.
