Let, the functional transform be,

Then the Laplace Transforms of the derivatives are respectively:

Take Laplace Transform of both sides.

Thus, we have, from the system of differencial equations,

Thus,

Thus, (note )

Substitute into second functional equation,

Multiply, to clear denominator,

Thus,

Thus,

Thus,

Thus,

Thus,

Thus,

Now I think there is something wrong with your problem because the procedure now is to decompose after factorization but the factorization is irrational, unlikely that they will give that as a homework problem.