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your answer looks fine. all you did was some math - really nothing to do with the laplace transforms.
the interesting part of the analysis is still has not been completed, though. you've been given a system with a voltage input that is a step function. the resultant answer you have is basically the response of the system, but in the s-domain.
to understand the current in the time domain, simplify the final answer you got, break it down into an equation showing the individual poles and zeroes, then use partial fraction expansion to get it into a form where all the terms in the denominator are by themselves in individual terms and then apply the reverse transform to get the time domain response.