The second-order KdV equation was introduced as a model to describe the wave propagation in a weakly nonlinear and weakly dispersive system where the existence of multi-soliton solution is raised as an open question. In this paper, we develop the procedures of the Hirota method for solving the equation and give a rigorous proof in the sense of symbolic computation that the equation can only admit the single solitary wave solution with no chance of obtaining the exact multi-soliton solution. With the effort and results illustrated, we hope it will trigger interests from all disciplines to search for the new type of multi-soliton solutions. (C) 2015 Elsevier Ltd. All rights reserved.