On the Reducibility of Certain Quadrinomials

In 2007 West Coast Number Theory conference Walsh asked to determine all irreducible polynomials of the form P (x) = x + x + x + 4 with integer exponents i > j > k > 0, such that for some positive integer l the polynomial P (x) is reducible in Z[x]. In this paper we prove that such polynomials are quadrinomials x4m + x3m + x2m + 4, where m is an odd positive integer. In addition, Walsh asked for the examples of reducible quadrinomials x +x +x +n, n > 4 with no linear or quadratic factors. We compute the examples of reducible polynomials of the form above with non-trivial factors and negative coefficient n.