There very well may be a sneaky, 'trick' answer to do this (like making use of the fact that det(AB)=det(A)det(B), but I don't see what two matrices would multiply to give you this)

If you're stuck using Sarrus, at the very least make the substitution x=n+2. That will balance out your terms a bit more, and at least give you some difference of squares while you're multiplying it all out.