Let say we have two polynomials of a integral domain $R[X]$. Let's call them $P$ and $Q$.
We suppose $\deg P + \deg Q = N$, with $N>1$ being an integer.
We also suppose that $Q$ is a second degree polynomial of the form $(X^2-m)$ , so that $Q$ is irreducible.
As $R[X]$ is integral , $\deg PQ=\deg P + \deg Q = N$.
Is there a way of knowing if $PQ$ has roots for $N$ fixed? I'm only interested about the result of existence.