A previously proposed self-consistent reptation model that includes chain stretching, chain-length fluctuations, segment connectivity, and constraint release is used to predict transient and steady shearing flows. Quantitative comparisons are made with the concentrated solution data considered in the previous papers of the series. The model is able to capture quantitatively all features of experimental data considered, including overshoot in both shear and first normal stresses, the strain-rate dependence of the strain magnitude at maximum stress, the steady-state viscosity and first-normal-stress coefficient as functions of shear rate, the viscosity curves for different molecular weight, the transient and steady-state behavior of the extinction angle, and the stress relaxation in cessation of steady shear flow. The model can describe all aspects of the data very well except the magnitude of the overshoot in stress at high shear rates, when the model is somewhat over-predictive. A new method of analysis for shear stress decay following cessation of steady shear is proposed, based on the physics of the model.