What can I say about $x^4 \equiv -4 \mod p$ where $p$ is prime?

Question

What can I say about $x^4 \equiv -4 \mod p$ where $p$ is prime? In general what can I do with powers that are greater than $2$ and where I cannot use reciprocity, legendre/jacobi etc... In general what can I say about a quadratic polynomial modulo $p$: For instance $(x-1)^2 \equiv 1 \mod p$

By 'what can I say' I mean $p \equiv$ something $\mod 4$ or $8$

Answer 1

$$ x^4 + 4 = ((x-1)^2 + 1) ((x+1)^2 + 1) $$