Let a, b,c, d be four integers (not necessarily distinct) in the set {1,2,3,4,5}. the no. of polynomials x^4+ax^3+bx^2+cx+d which is divisible by x+1 is

By either long or synthetic division, it's easy to show that



The quartic will be exactly divisible by when the numerator of the remainder term vanishes, or for those values of such that



I'm not sure how to count the number of solutions (software tells me it should be 80), but hopefully this is a helpful push in the right direction.