In this paper we propose a class of 8th-order potential function, and discuss the bifurcation characteristic of such a system in detail. Then, a symmetric quad-stable system consisting of two small-scale bistable potentials on the left and right and an intermediate barrier is obtained. In order to analyze the quad-stable system characteristic effectively, under the combined action of periodic force and random force, the approximate analytical expression of the quad-stable system output response is established. Meanwhile, from the viewpoint of the energy, the work which is a process quantity is introduced to describe the capacity for work between the large-scale and small-scale bistable potential. It is found that the double stochastic resonance phenomenon does exist in the quad-stable system. The theoretical analysis and numerical simulation results indicate that when the height of the intermediate barrier is higher than the barrier height of the two small-scale bistable potentials on the left and right, as the noise intensity increases, the response of the quad-stable system transforms a small-amplitude vibration restricted in a small-scale bistable subsystem into a large-amplitude vibration across the intermediate barrier, and the work done by the periodic force presents a double-peak curve. To be more specific, as the noise intensity gradually increases from zero, the system response is first confined to a small-scale bistable potential. Under the joint action of the small-scale bistable potential, periodic force and random force, the small-scale stochastic resonance phenomenon occurs, and the first resonance peak appears. With the noise intensity increasing even further, the system response turns into the large-amplitude vibration between two small-scale bistable subsystems, resulting in the large-scale stochastic resonance phenomenon and a higher resonance peak. Thus, the work done by periodic force has the peak values at two different noise intensities, which means that the noise can induce the double stochastic resonance phenomenon in the quad-stable system. More importantly, it can be found that the small-scale stochastic resonance can enhance the effect of large-scale stochastic resonance.