We discuss N-soliton solutions and analyze the Hirota N-soliton
conditions, starting from Hirota bilinear forms. An algorithm to verify
the Hirota conditions is proposed by factoring out common factors for
the terms in the conditions and comparing degrees of the involved
polynomials containing the common factors. Applications to a class of
generalized KdV equations and a class of generalized higher-order KdV
equations are made, together with all proofs of the existence of
N-soliton solutions to each equation in two classes.