I came across the following problem:

What are conditions such that the polynomial $x^2+1$ divides $p(y)+q(z)+ax+b=F(x,\,y,\,z)$,?

Here $p$ and $q$ are also polynomials and $a$, $b$ are real numbers. The main difficulty is that $F(x,\, y, \,z)$ has three variables, and the idea of using roots cannot apply here. I am expecting a relation between $x$ and $y,z$, but I not able to find it.