We consider nonlinear nonhomogeneous Dirichlet problems driven by the sum of a p-Laplacian and a Laplacian. The hypotheses on the reaction term incorporate problems resonant at both +/- a and zero. We consider both cases p > 2 and 1 < p < 2 (singular case) and we prove four multiplicity theorems producing three or four nontrivial solutions. For the case p > 2 we provide precise sign information for all the solutions. Our approach uses critical point theory, truncation and comparison techniques, Morse theory and the Lyapunoff-Schmidt reduction method.