A system of nth-order differential equations with polynomial right-hand sides is considered, and a simple analytical method of estimating the domain of attraction is developed. The method involves an ordinary quadratic Lyapunov function v = X(T)(Y)X with a certain parameter vector Y from the same state space. All linear factors in the expression for v are bounded in the domain v less than or equal to 1, and the derivative is bounded by a quadratic function, the negativeness of which determines the restrictions for Y. The domain of attraction is estimated through a simple scaling of the obtained area or through its nonlinear transformation with optimization. The method allows for obtaining domains (for example, with infinite volume) that are comparable with ones obtained by complicated computational procedures. A set of examples is presented.