Let $f(x)$ and $g(x)$ be two relatively prime polynomials having integer coefficients with $g(0)\neq 0$. The authors show that there is an $N=N(f,g)$ such that if $n \geq N$, then the non-reciprocal part of the polynomial $f(x)x^{n}+g(x)$ is either irreducible or identically 1 or $-1$ with certain clear exceptions that arise from a theorem of Capelli. A version of this result is originally due to Andrzej Schinzel. The present paper gives a new approach that allows for an improved estimate on the value of $N$.