One of the central subjects in modern control engineering is the investigation of stability for arbitrary given nonlinear systems. In this contribution we will introduce a novel method to compute the domain of attraction (DA) of a polynomial system using bezoutians. The presented method is able to estimate the DA for a given system linear as well as nonlinear system. It works for quadratic Lyapunov function as well as for arbitrary non linear Lyapunov functions. In this present algorithm the DA is succesfully determined by an inner and outer bound. The inner bound yields an enclosure of the Lyapunov function (LF) countour line c(star).