>>13555238You forgot the coefficients on the leading terms in the factors. In any case, the constant term constrains the constant factors. The first degree terms depend on the constant factors and can be deduced pretty easily from those. It gets messier the higher the degree you get (since as m increases, the number of k-partitions of m increase for all 0 < k <= m. This is a fancy way of saying there are more ways for the degrees to add to m as m gets larger), but for degree 4, you should be able to brute force it easily with this method by starting with the constant terms and working your way up.
