Let \(f(X),g(X)\in{\mathbb Q}[X]\) be two relatively prime polynomials, with \(\deg(f) b\). The aim of the present paper is to extend these results to the case of polynomials of the form \(n_1f(X)+n_2g(X)\) where, as before, \(f(X),g(X)\in{\mathbb Q}[X]\) are relatively prime, \(n_1,n_2\) are integers, and \(\deg(f)\leq \deg(g)\). Let \(\Omega(n)\) be the number of prime divisors of \(n\), counted with multiplicity, and, similarly, let \(\Omega(h(X))\) be the number of irreducible factors of the rational polynomial \(h(X)\), counted with multiplicity. For the case when \(\deg(f)<\deg(g)\), the main statement is the following: ``Let \(d\) be a positive divisor of \(n_2\); if \(| n_2/n_1| \) is greater than an explicit bound \(b=b(f,g,d)\), then \(\Omega(n_1f(X)+n_2g(X)) \leq \Omega(n_2/d)\)''. Specializing to the case \(n_1=1\) and \(n_2=p\) prime, the author improves on the previous bound \(b\). A similar statement is proved for the case when \(\deg(f)=\deg(g)\).