I am able to solve this question in full, but I have a question about the hint. I need help justifying why the Hint implies that $P$ is prime. Thanks.
Here is what I have so far.
Let $f,g\in R\setminus P$ where $fg\in P$. We need to reach a contradiction. I am trying to apply the hint with $A=\langle P\cup \{f\}\rangle$ and $B=\langle P\cup \{g\}\rangle$. I'm having trouble showing that $AB\subseteq P$.
The problem is, We have something like $rfsgt\in AB$ with $r,s,t\in R$, which may not be in $\langle fg \rangle$. Should I have chosen $A,B$ differently?
Note that $R$ is assumed to have unity but may not be commutative.