A case of polynomial irreducibility

Question

Assume we have two polynomials $p(x),q(x)\in\mathbb{Z}[x]$ irreducible in $\mathbb{Q}[x]$ having degrees $m, n$ respectively. I am trying to prove that $p(x)*q(x)+ x^{m+n}$ is irreducible. Or it is not? In such case, do you have any idea how to make from $p$ and $q$ an irreducible polynomial?

Answer 1

Even with $p,q$ not associated: take $p(x)=x$, $q(x)=x+1$. Then $$x(x+1)+x^2=x(2x+1)$$ is factored.

Answer 2

Not true.

Take $q=-p$. Then $x^{2n}-p(x)^2 = (x^n+p(x))(x^n-p(x))$ is reducible.