I am trying to find the inverse Laplace transform of $$\frac{4s^3 + s}{s^2+1}$$
I tried polynomial long division and reduced it to the following expression:
$$4s - \frac{3s}{s^2+1}$$
But I'm not sure if I can do anything with this, since I haven't been taught (or even know if it's possible) to take the inverse Laplace of just $s$.
What else can I do here?
The inverse Laplace transform of $s$ is $\delta'$ (the derivative of Dirac's delta "function"). To understand what that means, you need to know a little about distribution theory.
(There is no way around this: if in your rational function $\frac{P}{Q}$, you have that $\deg P \ge \deg Q$, the inverse transform will contain $\delta$:s and their derivatives.)