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Source. Theorem 1, printed p. 4 (PDF p. 5) of the 22 May 2001 author manuscript.

Statement

For each positive integer $d$ with $4\mid d$, some polynomial

$$f(x)\in\mathbb Z^+[x]$$

makes $f(x)x^n+d$ reducible for every integer $n\geq0$; reducibility is over the rationals, as the paper's problem is posed (p. 1). Here the source uses $\mathbb Z^+[x]$ for polynomials with positive integral coefficients.

The positive-coefficient condition replaces the usual nontriviality condition $f(1)\neq-d$ in the source's formulation.

Proof pointer. Printed pp. 12--13 (PDF pp. 13--14) apply Theorem 3 to a system built from the repeated-modulus covering of Theorem 4. Its proof was not reconstructed or independently checked here.