Wilson's Theorem and quadratic Congruence $x^2 + 1 \equiv 0 \pmod p$

Question

Using Wilson's theorem, we show that if $p$ is an odd prime $x^2 + 1 \equiv 0 \pmod p$ has a solution if and only if $p\equiv 1\pmod 4$ . Why cant $p$ be of the form $4n+3$ ?