Long strings of composite values of polynomials and a basis of order 2

We show that for any polynomial f: Z→ Z with positive leading coefficient and irreducible over Q, if N is large enough then there are two strings of consecutive positive integers I₁=₁-m,…, n₁+m\ and I₂=₂-m, …, n₂+m\, where m = [(log N) (log log N)1/325525], such that I₁∪ I₂ ⊂ [1, N], N = n₁ + n₂, and $f(n)$ is composite for any n∈ I₁∪ I₂. This extends the result in [5] which showed the same result but with $f(n)=n$.