  In this paper a generalized Rayleigh-Li\'enard oscillator is consider and
lower bounds for the number of limit cycles bifurcating from weak focus
equilibria and saddle connections are provided. By assuming some open
conditions on the parameters of the considered system the existence of up to
twelve limit cycles is provided. More precisely, the approach consists in
perform suitable changes in the sign of some specific parameters and apply
Poincar\'e-Bendixson Theorem for assure the existence of limit cycles. In
particular, the method for obtaining the limit cycles through the referred
approach is explicitly exhibited. The main techniques applied in this study are
the Lyapunov constants and the Melnikov method.
