solving a simple DE using the Laplace Transform - am i doing it right?

Hi all. I have been trying to relearn the laplace transform. I want to solve this second order ode

I know this can be easier done using the characteristic equation but I want to use the laplace tansform for learning purposes

here is my solution - is this right? have i done something stupid?

step 1 -

*here i am confused. With no initial conditions does the equation become*

then

step 2 - I factorise the quadratic and that becomes

*i am sure what i did here with partial fractions is wrong*

step 3 - take the inverse laplace transform

if i take the inverse laplace transform i get the result from the characteristic equation

are my steps correct?

thanks for any help