Irreducible values of polynomials

Does there exist a polynomial f(X) such that all polynomials
f(X), f(X)+1, f(X)+2, ..., f(X)+285 are irreducible? Clearly the answer depends on the field the coefficients are taken from. We will discuss a generalization of this problem (aka Schinzel's hypothesis H for polynomial rings), some recent results, and the connection with Hilbert's irreducibility theorem.