Answer
$S$ does not span $P_2$.
Work Step by Step
The set $S=\left\{1-x, 1-x^{2}, 3 x^{2}-2 x-1\right\}$ is not a basis for $P_2$ because $S$ does not span $P_2$. For example, consider the constant function $f(x)=1$ as a linear combination of the elements of $S$ as follows
$$1=a(1-x)+b(1-x^{2})+c(3 x^{2}-2 x-1).$$
Comparing the coefficients, we have
\begin{align*}
a+b-c&=1\\
-a-2c&=0\\
-b+3c&=0.
\end{align*}
It is easy to note that the above system is in-consistent, since we have $a=-2c$ and $b=3c$. Now, substituting into the first equation we get $0=1$ which is a contradiction and hence the system is in-consistent.
