Hints:
A set of functions $\{f_1, f_2, \dots, f_k\}$ is linearly independent on an interval $I$ if and only if there exists $a\in I $ such that their Wronskian $W(f_1, \dots, f_k)(a) \neq 0$.
Solution: We recall that a set of functions $\{f_1, f_2, \dots, f_k\}$ is linearly independent on an interval $I$ if and only if there exists $a\in I $ such that their Wronskian $W(f_1, \dots, f_k)(a) \neq 0$. To show that the functions \( e^{2x}, x e^{2x}, x^2 e^{2x} \) are linearly independent, we will show that the Wronskian of these functions are nonzero at some point. The Wronskian of the functions will be
\begin{align*}
W(f_1, f_2, f_3)(x) & =
\begin{vmatrix}
f_1(x) & f_2(x) & f_3(x) \\
f_1'(x) & f_2'(x) & f_3'(x) \\
f_1''(x) & f_2''(x) & f_3''(x) \\
\end{vmatrix} \\
& =
\begin{vmatrix}
e^{2x} & x e^{2x} & x^2 e^{2x} \\
2e^{2x} & (2x+1)e^{2x} & (2x^2+2x)e^{2x} \\
4e^{2x} & (4x+4)e^{2x} & (4x^2+8x+2)e^{2x}
\end{vmatrix}.
\end{align*}
Take $x = 0$, then we have
\begin{align*}
W(f_1, f_2, f_3)(0) & =
\begin{vmatrix}
1 & 0 & 0\\
2 & 1 & 0 \\
4 & 4 & 2
\end{vmatrix} = 2.
\end{align*}
Thus, the given functions are linearly independent.