I've been tasked with finding the pulse transfer function G(z) of the combination of the following equation \begin{equation}G_p(s) = {10\over (s+1)(s+2)(s+3)} \end{equation} and a zero order hold.

I started by combining the hold with the equation to obtain

\begin{equation} {10(1-e^{-sT})\over s(s+1)(s+2)(s+3)} \end{equation}

and then splitting this up to the equations

\begin{align} {1\over s(s+1)} \tag1\\ {1\over (s+2)(s+3)}\tag2 \\ {10(1-e^{-sT})}\tag3 \end{align}

finding the z transform of each one, and finally combining them together. I'm not sure if I've went about this in the right way, the answer I obtained seems very complicated and I can't find any similar examples. Can anyone who knows how to do this tell me if I'm going about this correctly and point me in the right direction?