Calculating pulse transfer function
First z-transform \$\small(1-e^{-sT})\$ \$\rightarrow\$ \$\small(1-z^{-1})=\large\frac{z-1}{z}\$. Then write the remaining Laplace expression in partial fractions:
$$\small 10\left(\frac{A}{s}+\frac{B}{s+1}+\frac{C}{s+2}+\frac{D}{s+3 }\right)$$

Z-transform the bracket term-by-term using standard Laplace/z transform tables, and combine with \$\large\frac{z-1}{z}\$.