Inverse Fourier transform using theorem

Question

The Fourier transform of $h(x)=\frac{4}{3+2x+x^2}$ is $\hat h(x)=2 \sqrt \pi e^{ix-\sqrt 2 |x|}$.

Now I should use the following statement to determine which function $f: \mathbb R \to \mathbb C \in L^1$ has $\hat f(x)=\frac{4}{3+2x+x^2}$ as its fourier transform:

Let $f \in C^1(\mathbb R)$ and $f, g:=f' \in L^1(\mathbb R).$ Then $\hat g(x)=ix\hat f(x)$

$\frac{4}{3+2x+x^2}=ix(-i\frac{4}{3+2x+x^2})$.

Let $\hat g(x):=\frac{4}{3+2x+x^2}, \hat f(x):=-i\frac{4}{3+2x+x^2} \Rightarrow$

$\hat g(x)=ix \hat f(x) \Rightarrow$ (I'm not sure about the following, because we use the statement in the other direction)

$g=f'=\hat {\hat f}'(-x)$

and I don't know how to continue

Answer 1

Hint

Since $\hat h\in L^1$, $$h(x)=\int_{\mathbb R}\hat h (\alpha )e^{2i\pi \alpha x}\,\mathrm d \alpha .$$ In particular (doing the substitution $\alpha =-\xi$) $$h(x)=\int_{\mathbb R}\hat h(-\xi )e^{-2i\pi \xi x}\,\mathrm d \xi. $$

I let you conclude.