  The Riemann-Hilbert (RH) problem is first developed to study the focusing
nonlinear Schr\"{o}dinger (NLS) equation with multiple high-order poles under
nonzero boundary conditions. Laurent expansion and Taylor series are employed
to replace the residues at the simple- and the second-poles. Further, the
solution of RH problem is transformed into a closed system of algebraic
equations, and the soliton solutions corresponding to the transmission
coefficient $1/s_{11}(z)$ with an $N$-order pole are obtained by solving the
algebraic system. Then, in a more general case, the transmission coefficient
with multiple high-order poles is studied, and the corresponding solutions are
obtained. In addition, for high-order pole, the propagation behavior of the
soliton solution corresponding to a third-order pole is given as example.
