Answer
$S$ is not a basis for $P_3$.
Work Step by Step
The set $S=\left\{t^{3}-1,2 t^{2}, t+3,5+2 t+2 t^{2}+t^{3}\right\}$ is not a basis for $P_3$. Indeed, since the vector $5+2 t+2 t^{2}+t^{3}$ can be written as a linear combination of the other vectors of $S$ as follows
$$5+2 t+2 t^{2}+t^{3}=(t^{3}-1)+(2 t^{2})+2(t+3) $$
Therefore, $S$ is not linearly independent set of vectors, and hence $S$ is not a basis for $P_3$.
