Abstract: The quad-curl term is an essential part in the resistive magnetohydrodynamic (MHD) equation and the fourth order inverse electromagnetic scattering problem which are both of great significance in science and engineering. It is desirable to develop efficient and practical numerical methods for the quad-curl problem. In this paper, we firstly present some new regularity results for the quad-curl problem on Lipschitz polyhedron domains, and then propose a mixed finite element method for solving the quad-curl problem. With a novel discrete Sobolev imbedding inequality for the piecewise polynomials, we obtain stability results and derive optimal error estimates based on a relatively low regularity assumption of the exact solution.