Answer
See below
Work Step by Step
Given:
$x_1(t)=\begin{bmatrix}
t^2\\
6-t+t^3\\
\end{bmatrix}$
$x_2(t)=\begin{bmatrix}
-3t^2\\
-18t+3t^2-3t^3
\end{bmatrix}$
Obtain
$W_{[x_1,x_2]}=\begin{vmatrix}
t^2 & -3t^2\\
6-t+t^3 & -18+3t-3t^2
\end{vmatrix}=(3t-3)t^4$
Assume that $t=3$ we have $W_{[x_1,x_2]}(3)=486 \ne 0$
Hence, $x_1(t)$ and $x_2(t)$ are linearly independent on $(-\infty,\infty)$
