Inverse Fourier transform of $\frac{\frac{2}{1+2i\pi f}-\frac{2}{4+2i\pi f}}{\frac{1}{1+2i\pi f}+\frac{1}{3+2i\pi f}}$
This looks correct; obviously you need to simplify to obtain a transfer function with polynomials as numerator and denominator. Partial fraction expansion then leads to a form that can be directly transformed to the time domain. The only thing that might be wrong is the minus sign between the two exponentials.