Therefore,
and the RHS factors as the difference of two squares.
The title really says it all. My research adviser showed me the problem: is not prime, apparently. He said that the proof is very short, elegant and elementary, but that he couldn't have seen it unless someone showed him. I also don't see it.
There is not a general way to do these types of problems, unfortunately. However, specifically can be solved using the most elementary technique of just manually checking for primes less than, say, 31 or so. That is how I start any of these problems, given that of course the vast majority of odd numbers are divisible by one of these primes.
For example: