Show that for all functions $f, g \in C^2(\mathbf{R})$ such that the functions $f, f', f''$ and $g$ have slow growth, we have
$$\int_{-\infty}^{+\infty} L(f)(x)\,g(x)\,\varphi(x)\,\mathrm{d}x = -\int_{-\infty}^{+\infty} f'(x)\,g'(x)\,\varphi(x)\,\mathrm{d}x,$$
where $L(f)(x) = f''(x) - x f'(x)$ and $\varphi(x) = \frac{1}{\sqrt{2\pi}}\mathrm{e}^{-x^2/2}$.