:::I think that I can fully clarify the case u+v=0: if u+v=0, then the numerator of w is zero. Otherwise, a small variation of the coefficients would provide monic equations with bounded coefficients, with arbitrarily large roots (in absolute value); this is impossible. A purely algebraic proof is possible, using Galois theory, but I should think harder to find such an algebraic proof. As the numerator of w is zero if and only if the associated depressed quartic is bi-quadratic, one may eliminate this case by considering separately the quartics whose depressed form is bi-quadratic.  
 :::If one want a truly general formula, one may choose another determination of the cubic root defining Q. If the three determinations give u+v=0, this implies that all the coefficients of the associate depressed cubic are zero (to be verified, but I am pretty sure), thus the quartic is .