The starting point is a paper by L. Boccardo, F. Murat, J.P. Puel, where it is considered the zero Dirichlet boundary value problems associated to nonlinear elliptic equaltions with quadratic dependence on the gradient and whose model may be: -\Delta u - \lambda c(x) u - \mu(x)|\nabla u|^2 = h(x), \, x\in \Omega, with c, h \in L^p(\Omega) \, (p > N/2), \, c\ge c_0 >0 and 0<\mu_1\le \mu(x) \le \mu_2. If \lambda < 0, the existence of an a priori L^\infty estimate for the solutions of a family of approximated problems becomes the key to prove the existence of solution. As it was proved by V. Ferone, F. Murat (and also in a paper by Porretta), this a priori estimate may fail for \lambda = 0. Our aim is to go beyond the case \lambda \le 0. Indeed, we improve jointly with L. Jeanjean, C. De Cosner and K. Tanaka the previous multiplicity result of L. Jeanjean, B. Sirakov (see also a paper by B. Abdellaoui, A. Dall'Aglio, I. Peral) for the case that \lambda \ge 0 is small and \mu is constant. In contrast with the constant case where it is possible to apply a suitable change of variables to reduce to a variational semilinear problem, when \mu(x) is not constant, we study the bifurcation form infinity of solutions of the problem to solve it. Continuing previous works for the case \lambda = 0, we also see in collaboration with L. Moreno that the above multiplicity result fails when a singular term at u=0 is added to the term \mu(x) |\nabla u|^2.