Answer
$$W=-2 x^{2} \mathrm{e}^{x} \mathrm{e}^{-x}-4 \mathrm{e}^{x} \mathrm{e}^{-x}.$$
Work Step by Step
The Wronskian of $\left\{x, x^{2}, e^{x}, e^{-x}\right\}$ is given by
$$W=\left|\begin{array}{cccc}{x} & {x^{2}} & {\mathrm{e}^{x}} & {\mathrm{e}^{-x}} \\ {1} & {2 x} & {\mathrm{e}^{x}} & {-\mathrm{e}^{-x}} \\ {0} & {2} & {\mathrm{e}^{x}} & {\mathrm{e}^{-x}} \\ {0} & {0} & {\mathrm{e}^{x}} & {-\mathrm{e}^{-x}}\end{array}\right|=-2 x^{2} \mathrm{e}^{x} \mathrm{e}^{-x}-4 \mathrm{e}^{x} \mathrm{e}^{-x}.$$
