Show that every maximal ideal $\displaystyle m \subset \mathbb{R}[x,y]$ is one of the following:

*$\displaystyle m=<x-a,y-b>$ for some $\displaystyle a,b \in \mathbb{R}$;

*$\displaystyle m=<y-rx-s,x^2+vx+w>$ for some $\displaystyle r,s,v,w \in \mathbb{R}$ with $\displaystyle v^2-4w<0$;

*$\displaystyle m=<x-t,y^2+vy+w>$ for some $\displaystyle t,v,w \in \mathbb{R}$ with $\displaystyle v^2-4w<0$.