On the distribution in the arithmetic progressions of reducible quadratic polynomials

By using Weil's estimate for Kloosterman sums, we obtain a result about the distribution of the sequence $n(n+2)$, beyond the classical level, when a bilinear form with support over pairs of prime moduli is considered. We also obtain an analogous result in the case of a trilinear form, but only by using the recent results for sums of Kloosterman sums of Deshouillers–Iwaniec. Furthermore the method is extended to consider general quadratic reducible polynomials.