<p>In this paper, we investigate a variable-coefficient extended Korteweg-de Vries equation with an external force in fluid mechanics and plasma dynamics. Through the Hirota method, we work out the <i>N</i>-soliton solutions under some variable-coefficient constraints, where <i>N</i> is the positive integer. Based on the <i>N</i>-soliton solutions, <i>H</i>th-order breather, hybrid and multi-pole solutions are derived, where <i>H</i> is another positive integer. Effects of the variable coefficients on those nonlinear waves are discussed graphically. We find that the variable coefficients lead to the emergence of some types of the nonlinear waves, that the external force affects the velocities and backgrounds of those nonlinear waves, and that the dispersive and dissipative coefficients affect the characteristic lines and velocities of those nonlinear waves.</p>