sl(2) construction of type A N-fold supersymmetry

N-fold supersymmetry is an extension of the ordinary supersymmetry in one-dimensional quantum mechanics. One of its major property is quasi-solvability, which means that energy eigenvalues can be obtained for a portion of the spectra. We show that recently found type A N-fold supersymmetry can be constructed by using sl(2) algebra, which provides a basis for the quasi-solvability. By this construction we find a condition for the type A N-fold supersymmetry which is less restrictive than the condition known previously. Several explicitly known models are also examined in the light of this construction.