Finding the Laplace transform of $f(t)=(1-t)(u_1(t)-u_3(t))$

Question

Given $f(t)=(1-t)(u_1(t)-u_3(t))$, I am supposed to find the Laplace transform of $F(s) = {\mathcal L}\left\lbrace f(t) \right\rbrace$. I am not sure how I would start this question, as I have gotten as far as to distribute the $(1-t)$ term into the $(u_1(t)-u_3(t))$. Any help would be appreciated!

Answer 1

Hint: Use the fact that \begin{align} \mathcal{L}\{u_c(t) f(t-c)\}(s) = e^{-cs}F(s) \ \ \text{ where } \ \ \mathcal{L}\{f(t)\}(s) = F(s). \end{align}

For instance \begin{align} \mathcal{L}\{u_1(t)(1-t)\} = e^{-s}\mathcal{L}\{-t\}(s) = -\frac{e^{-s}}{s^2}. \end{align}