Answer
$S$ is a basis for $P_2$.
Work Step by Step
Let $S$ be given by $$
S=\left\{1, t, 1+t^{2}\right\}.
$$
Consider the combination
$$a +bt+c(1+t^{2})=0, \quad a,b,c\in R.$$
Which yields the following system of equations
\begin{align*}
a+c&=0\\
b&=0\\
c&=0.
\end{align*}
The above system
the solution $a=b=c=0$.
Then, $S$ is linearly independent set of vectors. Since, $P_2$ has dimension $3$ then, by Theorem 4.12 $S$ is a basis for $P_2$.
