Answer
See answer below
Work Step by Step
We are given $S=\{(x,x^3): x \in R\}$
Assume that $S=\{ v \in R^2: v=(x,x^3), x \in R\}$
then take $v=(1,1)$ and $w=(3,27) \in S$
Thus $v+w=(1,1)+(3,27)=(4,28) $
but $4^3 =64 \ne 28 \\
\rightarrow v+w \notin S$
Hence $S$ is not a subspace of $R^2$
