Problems solving this parametric equation

Question

Working in the ring $$\mathbb Z_7[x]$$ with $$p(x)=x^4+1$$ $$q(x)=x^2+3x-k$$ I should be able to find the parameter $$k\in \mathbb Z_7[x]$$ that would satisfy the polynomial division $$p(x):q(x)$$ I was just able to find the zeros of the p(x) function (of which only x-1 is set in the ring) but I don't know if that is useful and how to proceed. Surfing the web I just managed to find examples where the k is in the dividend but inverting that method doesn't seem to work. Any help is appreciated.

Answer 1

over the real numbers we have $$ x^4 + 1 = (x^2 - x \sqrt 2 + 1)(x^2 + x \sqrt 2 + 1) $$ In the field of seven elements, $2$ is a square, namely $$ 3^2 \equiv 2 \pmod 7 \; , $$ $$ 4^2 \equiv 2 \pmod 7 \; . $$