Answer
$S$ does not span $P_2$.
Work Step by Step
The set $S=\left\{6 x-3,3 x^{2}, 1-2 x-x^{2}\right\}$ 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
$$1=a(6 x-3)+b(3 x^{2})+c(1-2 x-x^{2}).$$
Comparing the coefficients, we have
\begin{align*}
-3a+c&=1\\
6a-2c&=0\\
3b-c&=0.
\end{align*}
It is easy to note that the above system is in-consistent, since the first and the second equations give that $0=1$ which is a contradiction and hence the system is in-consistent.
