Problem: Show that $2(n-1)! \equiv -1 \mod n+2 \iff n+2$ is a prime.
I know that Wilson's theorem states that $(n-1)! \equiv -1 \mod p $ for $p$ a prime, so that is the important thing to know with these type of problems.
I know that $-1 \equiv n+1 \mod n+2$ , and that if $n+2$ is prime, then $(n+1)! = -1 \mod n+2$. So this is all I have for now, any hints appreciated.