Inverse Laplace transform of $F(s)=\frac{3s+7}{s^2-2s-3}$ I have to calculate the inverse Laplace Transform of this image:
$$F(s)=\frac{3s+7}{s^2-2s-3}$$
I try decomposing it in this way:
$$F(s)=\frac{3s-3+10}{(s-1)^2-4}=3\frac{s-1}{(s-1)^2-4}+5\frac{2}{(s-1)^2-4}$$
where I can identify that the original function is $$f(t)=3e^t\cosh(2t)+5e^t\sinh(2t)$$
But the textbooks says that the result should be: $f(t)=-e^{-t}+4e^{3t}$ and I can't find where my mistake is.

You are correct! Indeed, we have that
$$f(t)=3e^t\cosh(2t)+5e^t\sinh(2t)=3e^t\frac{e^{2t}+e^{-2t}}{2}+5e^t\frac{e^{2t}-e^{-2t}}{2}=4e^{3t}-e^{-t}.$$
