I came across the following problem:

How one can makes some conditions such that the polynomial $x^2+1$ divide $p(y)+q(z)+ax+b=F(x,y,z)$. Here $p$ and $q$ are some polynomials and $a$ and $b$ are real numbers. The main difficulty here is that $F(x,y,z)$ has three variables and the fact the idea of using roots cannot apply here. I am expected a relation between $x$ and $y,z$, but I not able to find it.