The stability and chaotic motions of a 3-D quadratic autonomous system with a four-wing chaotic attractor are investigated in this paper. Base on the linearization analysis, the stability of the equilibrium points is studied. By using the undetermined coefficient method, the heteroclinic orbits are found and the convergence of the series expansions of this type of orbits is proved. It analytically demonstrates that there exist heteroclinic orbits of Silnikov type connecting the equilibrium points. Therefore, Smale horseshoes and the horseshoe chaos occur for this system via the Silnikov criterion.