In this paper, a generalized variable-coefficient Calogero-Bogoyavlenskii-Schiff equation is investigated. Based on the Bell polynomials and an auxiliary variable, bilinear forms of such an equation are obtained. One-, two-, and three-soliton solutions are given through the Hirota method and symbolic computation. N-soliton solutions are also constructed. Multi-soliton interaction and propagation are investigated and illustrated: (i) properties of the multi-soliton interaction on different planes in space depend on the forms of the only variable coefficient; (ii) positions of the solitons change when the wave numbers have the reverse signs.