Answer
The vectors of $S$ are not linearly independent.
Work Step by Step
The set $S=\left\{1,2 x, x^{2}-4,5 x\right\}$ is not a basis for $P_2$ because the vectors in $S$ are not linearly independent. Indeed, one can see that the following linear combination
$$c_1+2c_2x+c_3(x^{2}-4)+5c_4x=0, \quad c_1,c_2,c_3,c_4\in R$$
is valid for the choice $c_1=c_3=0$ and any non-zero values of $c_2$ and $c_4$ such that $2c_2=-5c_4$, hence the vectors of $S$ are not linearly independent.
