We consider nonmonic quadratic polynomials acting on a general or on a finite-dimensional linear space as a continuation of our work in [
]. Conditions are given for the existence of right roots, if the coefficient operators have lower block triangular representations. In the finite-dimensional case we consider (in a certain sense, entrywise) nonnegative coefficient matrices in the general (reducible) case, and extend several earlier results from the case of irreducible coefficients. In particular, we generalize results of Gail, Hantler and Taylor [
]. We show that our general methods are sufficiently strong to prove a remarkable result by Butler, Johnson and Wolkowicz [
], proved there by ingenious ad hoc methods.