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In this article, we propose a new method to construct many new exact solutions with parameters for the generalized KdV–mKdV (GKdV–mKdV) equation with higher-order nonlinear terms. The proposed method is a generalization of the well-known G′G,1G-expansion method in the case of the hyperbolic function solutions. Also, we use a direct algebraic method based on the generalized Liénard equation to find other diffrent new exact solutions of the above GKdV–mKdV equation. Soliton solutions, periodic solutions, rational functions solutions, hyperbolic functions solutions and symmetrical hyperbolic Lucas functions solutions are obtained. Comparing our new results obtained in this article with the well-known results are given. The generalized G′G,1G-expansion method presented in this article is straightforward, concise and it can also be applied to other nonlinear partial differential equations in mathematical physics.