In this article, the mean-square filtering problem for polynomial system states over polynomial observations is studied proceeding from the general expression for the stochastic Ito differentials of the mean-square estimate and the error variance. In contrast to the previously obtained results, this article deals with the general case of nonlinear polynomial states and observations. As a result, the Ito differentials for the mean-square estimate and error variance corresponding to the stated filtering problem are first derived. The procedure for obtaining an approximate closed-form finite-dimensional system of the filtering equations for any polynomial state over observations with any polynomial drift is then established. In the example, the obtained closed-form filter is applied to solve the third-order sensor filtering problem for a quadratic state, assuming a conditionally Gaussian initial condition for the extended third-order state vector. The simulation results show that the designed filter yields a reliable and rapidly converging estimate.