  Let $K$ be an algebraically closed field and $\mathrm{M}(2,K)$ be the
$2\times 2$ matrix algebra over $K$ and $\mathrm{GL}(2,K)$ be the invertible
elements in $\mathrm{M}(2,K)$. We explore the image of polynomials with
constants, namely from the free algebra $\mathrm{M}(2,K)\langle x, y\rangle$.
In this article, we compute the images of the polynomial maps given by (a)
generalized sum of powers $Ax^{k_1} + By^{k_2}$ and (b) generalized commutator
map $Axy -Byx$, where $A$, $B$ are non-zero elements of $\mathrm{M}(2,K)$. We
compute this in the first case by fixing a simultaneous conjugate pair for $A,
B$ and it turns out that it is surjective in most of the cases. In the second
case, we show that the image of the map is always a vector space.
