A state-feedback synthesis problem for polynomial systems with guaranteed quadratic costs is addressed in this paper. Specifically, we focus on the local synthesis, that is, the closed-loop systems are assumed to be operated in a compact domain characterized by a semi-algebraic set. By means of rational Lyapunov functions, sufficient conditions for the synthesis problem can be formulated as state-dependent linear matrix inequalities (SDLMIs), which can be reformulated into standard LMIs using the sum-of-squares technique. Construction of alternative sufficient conditions with reduced computational cost for the synthesis problem is also discussed in this paper.