Non-homogeneous heat equation using the Fourier transform with a particular term

Question

I´m trying to solve $u_t(x,t) = u_{xx}(x,t) - xu_x(x,t), x \in R, t > 0,$ and $u(x,0) = f(x)$ using the Fourier transform. I have seen an example where instead of $xu_x$ they put a known function, but I don´t know what to do when $u$ is involved this way.