Determine whether $712! + 1$ is a prime number or not

Since $719$ is prime, by Wilson's theorem we have that $$718!\equiv 718\cdot 717\cdot 716\cdot 715\cdot 714\cdot 713\cdot 712!\equiv -1\pmod {719}$$

To prove that $719\mid 712!+1$, it is sufficient to prove that $$\begin{align}718\cdot 717\cdot 716\cdot 715\cdot 714\cdot 713&\equiv 1\pmod {719}\\\iff(-1)\cdot(-2)\cdot(-3)\cdot(-4)\cdot(-5)\cdot(-6)&\equiv 1\pmod {719}\\\iff 6!&\equiv 1\pmod {719}\\\iff 720&\equiv 1\pmod {719}\ \ \ \square \end{align}$$

No, $712!+1$ is not prime, see here in integer sequences. Further references are given there. The next factorial prime here is $872!+1$. See also the comments here, and this article by Caldwell and Gallot.