I need find $x^2\cdot e^{{-x^2}/2} * e^{{-x^2}/2}$. I used statements, that $\widehat{xf}=i \widehat{f}'$ and $\widehat{f*g}=\sqrt{2\pi}\widehat{f}\cdot \widehat{g}$. So, $\widehat{x^2\cdot e^{{-x^2}/2}}(\lambda)={e^{{{-\lambda^2}/2}} \cdot(1-\lambda^2)}$ and $\widehat{e^{{-x^2}/2}}(\lambda)=e^{-\lambda^2/2}$. Then, $\widehat{{(x^2\cdot e^{{-x^2}/2})*e^{{-x^2}/2}}}(\lambda)=\sqrt{2\pi}\cdot e^{-\lambda^2}\cdot(1-\lambda^2)$.
So, $(x^2\cdot e^{{-x^2}/2} * e^{{-x^2}/2})(y)= \int_{-\infty}^\infty e^{-\lambda^2+i\lambda y} \cdot (1-\lambda^2) \, d\lambda$, but I don't know how solve this integral, perhaps, I did some mistake in counting. Please help me solve this problem.