This paper describes synthesis of controllers involving Quadratic Programming (QP) optimization problems for control of nonlinear systems. The QP structure allows an implementation of the controller as a piecewise affine function, pre-computed offline, which is a technique extensively studied in the field of explicit model predictive control (EMPC). The nonlinear systems being controlled are assumed to be described by polynomial functions and the synthesis also generates a polynomial Lyapunov function for the closed-loop system involving the obtained controller. The synthesis is based on a sum-of-squares (SOS) stability verification for polynomial discrete-time systems, described in continuous-time in this paper. The presented synthesis method allows a design of EMPC controllers with closed-loop stability guarantees without relying on a terminal cost and/or constraint, and even without using the prediction horizon concept to formulate the control optimization problem. In particular, for a specified QP structure the method directly searches for the stabilizing coefficients in the cost and/or the constraint set. The method involves two phases, where the first searches for stabilizing controllers by minimizing a polynomial slack function introduced to the SOS stability condition and the second phase optimizes some user-specified performance criteria. The two phases are formulated as optimization problems which can be tackled by using a black-box optimization technique such as Bayesian optimization, which is used in this paper. The synthesis is demonstrated on a numerical example involving a bilinear model of a permanent magnet synchronous machine (PMSM), where in order to demonstrate the modeling flexibility of the proposed synthesis method a QP-based controller for speed regulation of PMSM is synthesized that is robust to parametric uncertainty coming from the temperature-dependent stator resistance of the PMSM.