How do I get the inverse laplace transform of $(s+2)*e^{-s}/(s+1)^2$

Question

There is no way to break apart this fraction I think, and I don't see a property on a Laplace transform table. How would I go about this?

Answer 1

Hint: write $$\frac{(s+1+1)e^{-s}}{(s+1)^2}=\frac{e^{-s}}{s+1}+\frac{e^{-s}}{(s+1)^2}$$ and use $${\cal L}(e^{ct})=\frac{1}{s-c}$$ $${\cal L}(t^ne^{ct})=\frac{n!}{(s-c)^{n+1}}$$ $${\cal L}\Big(H_c(t)f(t-c)\Big)=e^{cs}{\cal L}(f)$$ where $H(t)$ is Heaviside function.