Proof: For any prime number $p$, there must be integer $a$, $b$, $c$, $d$ such that $x^4+1 \equiv (x^2+ax+b)(x^2+cx+d)\ (mod\ p)$.

Question

This is a question comes from a book about Number theory, and the question is related to Quadratic residue.

I start with

$$ x^4+1 \equiv (x^2+ax+b)(x^2+cx+d) \\ \equiv x^4+(a+c)x^3+(b+d+ac)x^2+(ad+bc)x+bd\ (mod\ p) $$

and then

$$ \begin{cases} a+c \equiv b+d+ac \equiv ad+bc \equiv 0\ (mod\ p)\\ bd \equiv 1\ (mod\ p) \end{cases} $$

and I don't know what to do next.

The Legendre symbol and Jacobi symbol Properties and Calculating is ok for me.