Effects of the quintic nonlinearity for the ultrashort optical pulse propagation in a non-Kerr medium, or in the twin-core nonlinear optical fiber or waveguide, can be described by the coupled cubic-quintic nonlinear Schrodinger equations with variable coefficients. For such equations, a Lax pair and the infinitely-many conservation laws are constructed. Under certain variable-coefficient constraints, bilinear forms, bilinear Backlund transformations, bright/dark one-, two- and N-soliton solutions are derived via the Hirota method and symbolic computation. Those soliton solutions are merely related to the delayed nonlinear response effect, b(z). Graphical analyses on the soliton propagation and interaction suggest that b(z) can affect the bright/dark-soliton velocities but has no effect on their amplitudes. With the different choices of b(z), propagation of the linear-, periodic-, parabolic- and S-type bight and dark one solitons are seen, and different types of the interaction between the bright two solitons are displayed, such as the interactions between the bidirectional bright two ones, between the unidirectional bright two ones and between a moving bright one and a stationary one. Interactions between the linear-, S-, and parabolic-type dark two solitons are asymptotically analyzed and graphically illustrated as well, and those interactions are all elastic.