[tex]f(t)=\begin{cases}0&\text{for }t<2\\t^2-4t+8&\text{for }t\ge2\end{cases}[/tex]
[tex]\mathcal L_s\{f(t)\}=\displaystyle\int_0^\infty f(t)e^{-st}\,\mathrm dt[/tex]
[tex]=\displaystyle\int_2^\infty(t^2-4t+8)e^{-st}\,\mathrm dt=\frac{2e^{-2s}}{s^3}+\frac{4e^{-2s}}s[/tex]
