The numerical treatment of large-scale, nonsymmetric algebraic Riccati equations (NAREs) by a low-rank variant of Newton\'s method is considered. We discuss a method to compute approximations to the solution of the NARE in a factorized form of low rank. The occurring large-scale Sylvester equations are dealt with using the factored alternating direction implicit iteration (fADI). Several performance enhancing strategies available for the factored ADI as well as the related Newton-ADI for symmetric algebraic Riccati equations are generalized to this combination. This includes the efficient computation of the norm of the residual matrix, adapted shift parameters strategies for fADI, and an acceleration of the Newton\'s scheme by means of a Galerkin projection. Numerical experiments illustrate the capabilities of the proposed method to solve high-dimensional NAREs.