An irreducibility criterion for the sum of two relatively prime polynomials

We partly extend a result of Cavachi and Bonciocat on the sum of two relatively prime polynomials and prove that a polynomial of the form $f(X) + Ng(X)$, where $f(X), g(X) \in \mathbb{Z}[X]$ are two non-zero relatively prime polynomials with $\deg f ＜ \frac{1}{2}\deg g$, is irreducible over $\mathbb{Q}$ for all but finitely many square-free positive integers $N$. In addition, we derive a necessary and sufficient condition for a polynomial $r + p^{2}g(X) \in \mathbb{Z}[X]$ to be reducible over $\mathbb{Q}$ for a sufficiently large prime number $p$.