Let (Formula presented.) be the (Formula presented.) matrix algebra over (Formula presented.) and (Formula presented.) be the invertible elements in (Formula presented.). Inspired by Kaplansky–Lv́ov conjecture, we explore the image of polynomials with constants, namely polynomials from the free algebra (Formula presented.). In this article, we compute the images of the polynomial maps given by (a) generalized sum of powers (Formula presented.) and (b) generalized commutator map (Formula presented.), where (Formula presented.), (Formula presented.) are nonzero elements of (Formula presented.) when (Formula presented.) is an algebraically closed field. We show that the images of these maps are vector spaces. For the polynomial in (a), we compute the images by fixing a simultaneous conjugate pair for (Formula presented.), and it turns out that it is surjective in most cases.