$ \mathcal{X}(\omega) = \frac{\frac{1}{2j}}{(2 - j4 + jw)^{2}} - \frac{\frac{1}{2j}}{(2 + j4 - jw)^{2}} $
$ \ x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{j\omega t}\,d\omega $
$ = \frac{1}{2\pi}\int_{-\infty}^{\infty}(\frac{\frac{1}{2j}}{(2 - j4 + jw)^{2}} - \frac{\frac{1}{2j}}{(2 + j4 - jw)^{2}})e^{j\omega t}\,d\omega $
by looking at the table on p. 329 of the book some observations can be made:
the form $ \frac{1}{(2 - j4 + jw)^{2}} $ means that there is a term of the form $ \ te^{-2t} $
