Differential Equations Inverse Laplace(Partial Fractions)

[Econometricia]

1. L^-1{(3s+2)/ (s²+2s +10)}
After completing the square I get to 3s+2 /(s+1)² + 3² which suggests two solutions or one. They decompose the fraction into [(A)s+1 /(s+1)² + 3² ]+ [(B) 3/(s+1)² + 3²]
I am unsure of how this decomposition works I thought that we would take A(3s) as the numerator and B(2) as the other. If some one can clarify It would be much appreciated =)

 

[rock.freak667]

From here

(3s+2) /[(s+1)² + 3²]

Split this as

3s/[(s+1)² + 3²] + 2/[(s+1)² + 3²]

Now remember that Laplace transforms for cos(kt) and sin(kt), and apply the shift theorem. No need for partial fractions here.