Laplace Transform, Unit Step Function

Question

Hi I have been trying to do this Laplace Transform and cant seem to figure it out and was wondering if someone could point me in the right direction; here it is:

$$L_t{(u(t-2)(2t^2-6t+5)})$$

What I tried was:

$$L_t{(u(t-2)(2t^2-6t+5)})$$

$$=e^{-2s}L_t({2t^3-10t^2+17t-10})$$

$$=e^{-2s}(\cfrac{12}{s^4} - \cfrac{20}{s^3} + \cfrac{17}{s^2} - \cfrac{10}{s})$$

But this is wrong. Can anyone tell what I have done wrong and help me figure it out.?

Thanks.

Answer 1

By the formula $L(u(t-a)f(t))=e^{-as}Lf(t+a)$ we have
\begin{aligned} L(U(t-2)(2t^2-6t+5)) &=e^{-2s}L((2(t+2)^2-6(t+2)+5)) \\ &=e^{-2s}(\frac{4}{s^3}+\frac{2}{s^3}+\frac{1}{s}) \\ \end{aligned}