Find the Laplace transform of the following hard equation

Question

Ok so the objective is to factor this into something that resembles the Laplace tables. Give me some help pls. thx

Answer 1

HINT:

Using a little variant Partial Fraction Decomposition,

$$\frac{4s+1}{s(2s^2+2s+1)}=\frac As+\frac{B\left(s+\dfrac12\right)+C}{2s^2+2s+1}$$

as $$\frac1{2s^2+2s+1}=\frac12\cdot\frac1{\left(s+\dfrac12\right)^2+\left(\dfrac12\right)^2}$$