The problem sounds like this.
Show that $s\to\int_\mathbb{R}e^{-(x+is)^2}dx$ is constant wrt $s\in\mathbb{R}.$ Then use this fact to shot that $\mathcal{F}(e^{-a|x|^2})=e^{-\frac{|x|^2}{a}}$ for $a>0,$ where $\mathcal{F}$ is the Fourier transform on $\mathbb{R}$
I know the traditional ODE way to show this fact about the Gaussian bell but this proof got me interested and I don't have any clue on solving this. I've tried to differentiate wrt $s$ but no luck.