A coprimality condition on consecutive values of polynomials

Let ∈ℤ[] be a quadratic or cubic polynomial. We prove that there exists an integer ⩾2 such that for every integer ⩾ one can find infinitely many integers ⩾0 with the property that none of (+1),(+2),⋯,(+) is coprime to all the others. This extends previous results on linear polynomials and, in particular, on consecutive integers