1. The problem statement, all variables and given/known data Factor x^3+3x+6 over Z5, Z10, Z, Q and ℝ. 2. The attempt at a solution For Z5, I have the roots of x^3+3x+1 in Z5, x=2=-3 and x=3=-4, so x+3 and x+4 are factors. By long division of x^3+3x+1, it is found that x+3 is a repeated factor, so x^3+3x+6=(x+4)(x+3)^2. For Z and Q, Eisenhart's criterion is satisfied with 3, hence x^3+3x+6 is irreducible over Q, and hence Z. For Z10 and ℝ... I don't really get how to do this problem. Is there a consistent way to work out the factors for Z10? It seems like a lot of effort. Also, I noticed that x^3+3x+6 has one real root over ℝ, and so I can factor it neatly to a linear factor (with real root) and a quadratic factor with two complex roots, but I only managed to get these roots with a computational method - and they have pretty long but explicit expressions, so again, my question: is there a proper way to work out the roots by hand here? Thanks!