Answer
See answer below
Work Step by Step
We are given $S=\{(x,y): x^2-y=0\}$
Take the vectors $v=(1,1)$ and $w=(3,9)$ in $R^2$
then $1^2-1=0\\
3^2-9=9-9=0 \\
\rightarrow v,w \in S$
Thus $v+w=(1,1)+(3,9)=(4,10) \\
4^2-10=6 \ne 0 \\
\rightarrow v+w \notin S$
Hence $S$ is not a subspace of $R^2$
