Under investigation in this paper is a fourth-order variable-coefficient nonlinear Schrodinger equation, which can describe an inhomogeneous one-dimensional anisotropic Heisenberg ferromagnetic spin chain or alpha helical protein. With the aid of the Hirota method and symbolic computation, bilinear forms, dark one-and two-soliton solutions are obtained. Influences of the variable coefficients on the dark one and two solitons are graphically shown and discussed. Amplitude and shape of the dark one soliton keep invariant during the propagation when the variable coefficients are chosen as the constants. With the variable coefficients being the functions, amplitude of the dark soliton keeps unchanged during the propagation, but direction of the soliton curves. Head-on and overtaking collisions between the dark two solitons are displayed with the variable coefficients chosen as the constants, and it is shown that the shapes of the two solitons do not change during the collision. When we choose the variable coefficients as the functions, directions of the two solitons change and elastic collisions occur between the two solitons.