My question concerns the sums of two reducible polynomials with bounded coefficients. In particular, does there exist an absolute constant $C > 0$ such that for any two non-proportional reducible polynomials $f,g \in \mathbb{Z}[x]$ of the same degree $d \geq 2$ there exist integers $u,v$ such that $|u|, |v| \leq C$ and $uf(x) + vg(x)$ is irreducible?

My question concerns the sums of two reducible polynomials with bounded coefficients. In particular, for every $d \geq 2$ does there exist a number $C(d) > 0$ such that for any two co-prime reducible polynomials $f,g \in \mathbb{Z}[x]$ of degree $d$ there exist integers $u,v$ such that $|u|, |v| \leq C(d)$ and $uf(x) + vg(x)$ is irreducible?