The polynomial $x^4$ is an element of $P_4$. Can we write this element as a linear combination of the elements of $T$? To wit, are there scalars $a_1$, $a_2$, $a_3$ such that
\begin{align*}
x^4&=a_1\left(x^2-x+5\right)+a_2\left(4x^3-x^2+5x\right)+a_3\left(3x+2\right)
\end{align*}
Massaging the right side of this equation, according to the definitions of Example VSP, and then equating coefficients, leads to an inconsistent system of equations (check this!). As such, $T$ is not a spanning set for $P_4$.