I am working on the polynomial ring with two variables $\mathbb{R}[x,y]$ and trying to find the kernel and image of the map below $$\alpha: \langle y \rangle^{r+1} \oplus \langle x \rangle^{r+1} \oplus \langle y \rangle^{r+1} \oplus \langle x \rangle^{r+1} \to \langle x,y \rangle^{r+1}$$ defined by $\alpha(f,g,h,t) = -f-g-h-t$.
I think $\operatorname{Im} \alpha = \langle y \rangle^{r+1} \oplus \langle x \rangle^{r+1}$ but I need some help about the kernel. Thanks.