<p>In this paper, we investigate a generalized variable-coefficient extended Korteweg-de Vries equation with an external-force term for a fluid or a cosmic plasma. We obtain a Lax pair under some constraints via the Abolwitz-Kaup-Newell-Segur procedure. Based on that Lax pair, infinite conservation laws, Riccati- and Wahlquist-Estabrook-type auto-Bäcklund transformations are constructed. Via the Hirota method, we obtain some bilinear forms and <i>N</i>-soliton solutions, and derive the <i>H</i>th-order breather and hybrid solutions through the complex conjugated transformations, where <i>N</i> and <i>H</i> are two positive integers. The higher-order smooth positon solutions are constructed through the limit method. Moreover, those solutions are graphically presented under some choices of the variable coefficients and parameters.</p>