Using an extended quadratic Lyapunov function of the form V(x)=x^TP(x)x, we derive a sufficient condition that an input-affine polynomial-type nonlinear system is internal stable and the L₂-gain is less than or equal to some given positive constant. The condition is given as Riccati-type inequality, for P(x), which depends on the state. We can obtain the solution P(x) as a polynomial function by solving linear matrix inequalities, and determine the region of internal stability. We also illustrate that the proposed method is effective through a numerical example of bilinear systems.