For

you don't need to use PFD write s^2 + 8 as s^2 + 4 + 4 and separate

Write as [(s^2+4)/[(s^2+4)(s^2+9)] + 4/[(s^2+4)(s^2+9)]

L(s) = 1/(s^2+9) + 4/(s^2+4)(s^2+9) both of which are standard transforms

y(t) = 1/3sin(3t) + 4[-1/5*(1/3sin(3t)-1/2sin(2t)]

= 1/3sin(3t)-4/15sin(3t) + 2/5sin(2t)

= 1/15sin(3t) + 2/5sin(2t)