Let $p,q \in \mathbb{Q}[x]$ two relatively prime polynomials. Let $h\in \mathbb{R}$ any number and let $F_h(x) = p(x) + h \cdot q(x)$.

What can be said about the irreducibility of the polynomial $F_h(x)$ over the field $\mathbb{Q}(h)$?

More precisely, let $H = \{h : F_h \mbox{ is reducible over }\mathbb{Q}(h) \}$. How big can $H$ be?

Is there some reasonable condition on $p$ and $q$ that guarantees that $H$ is empty? Or finite? Or at least discrete in some sense?

I understand that $H$ can only contain algebraic numbers. The linearity of $F$ in $h$ forces $F$ to be irreducible for transcendental $h$, because one of the factors would have to lie in $\mathbb{Q}(x)$ and that would contradict that $p$ and $q$ are coprime.

I would guess that the same is true when $h$ is algebraic but its degree is large relative to the degrees of $p$ and $q$, though I don't know how to approach this.