We present a new analytic scheme for the description of the phase behavior of solutions of lattice polymers and hard particles. The theory covers mixtures containing short polymers (i.e., the radius of gyration smaller than the hard particle radius) as well as mixtures containing very long polymers. The lattice polymers have excluded volume interactions, and the hard particles can have a spherical, convex, or dumbbell shape. The theory combines the Flory theory for polymer solutions, the Boublik-Nezbeda equation of state for isotropic fluids of hard particles, and a simple correction term for the reduction of polymer configurations near a wall. In order to be able to integrate these different approaches, typical properties of the hard particles, such as the average number of surface-fluid contacts of a particle, are expressed in terms of lattice sites using a simple numerical simulation. These simulation results are fitted for general use by simple second-order polynomials. The resulting equation of state predicts entropy-driven as well as energy-driven demixing and restabilization. As examples, the ternary mixtures lattice polymer + solvent + (hard spheres or hard dumbbells) are discussed, as well as the comparison with the Flory theory for two different polymers and a solvent.