Answer
$$W=-4 \mathrm{e}^{x^{2}} \mathrm{e}^{x} x^{6}+2 \mathrm{e}^{x^{2}} \mathrm{e}^{x} x^{5}+6 \mathrm{e}^{x^{2}} \mathrm{e}^{x} x^{4}-2 \mathrm{e}^{x} x^{3} \mathrm{e}^{x^{2}}-6 \mathrm{e}^{x^{2}} \mathrm{e}^{x} x^{2}.$$
Work Step by Step
The Wronskian of $\left\{x^{2}, e^{x^{2}}, x^{2} e^{x}\right\}$ is given by
$$W=\left[\begin{array}{lll}{x^{2}} & {\mathrm{e}^{x^{2}}} & {\mathrm{e}^{x} x^{2}} \\ {2 x} & {2 x \mathrm{e}^{x^{2}}} & {\mathrm{e}^{x} x^{2}+2 x \mathrm{e}^{x}} \\ {2} & {2 \mathrm{e}^{x^{2}}+4 x^{2} \mathrm{e}^{x^{2}} }&{ \mathrm{e}^{x} x^{2}+4 x \mathrm{e}^{x}+2 \mathrm{e}^{x}}\end{array}\right|\\=-4 \mathrm{e}^{x^{2}} \mathrm{e}^{x} x^{6}+2 \mathrm{e}^{x^{2}} \mathrm{e}^{x} x^{5}+6 \mathrm{e}^{x^{2}} \mathrm{e}^{x} x^{4}-2 \mathrm{e}^{x} x^{3} \mathrm{e}^{x^{2}}-6 \mathrm{e}^{x^{2}} \mathrm{e}^{x} x^{2}.$$
