We develop the Riemann-Hilbert (RH) method for the generalized inhomogeneous Hirota equation with zero boundary condition. The RH problem is related to two kinds of scattering data: N simple poles and one N-order pole. Here we consider that when the scattering data have one or more higher-order poles, the formulas of bound-state (BS) solitons and multiple BS solitons are obtained, and the interaction between solitons and BS solitons are shown.