There are several well-known criteria for a polynomial with integer coefficients to irreducible over $\mathbb{Z}$, for example, Eisenstein's criterion. I'm looking for the opposite: is there some sufficient condition to show that a given polynomial must be reducible (apart from demonstrating a factorization)? Ideally I'd like some property that depends only on the coefficients appearing in the polynomial, like in Eisenstein's criterion.

There are several well-known criteria for a polynomial with integer coefficients to be irreducible over $\mathbb{Z}$, e.g., Eisenstein's criterion. I'm looking for the opposite: other than factorization, is there some sufficient condition to show that a given polynomial must be reducible? Ideally, I'd like some property that depends only on the coefficients appearing in the polynomial, like in Eisenstein's criterion.