How can this problem be solved with partial fractions? (Zero order holder with g(s) block function)
(I tried to follow the things on the whiteboard but I couldn't get what is going on, but this is how you would get partial fractions once you find your system \$H(z)\$)

When you have a system with many poles such as 
$$ H(s) = \frac{s+d}{(s+a)(s+b)} $$
or a discrete one such as
$$ H(z) = \frac{z+d}{(z+a)(z+b)} $$
You can apply the Heaviside method to get the partial fraction expansion.
With it, you want to change
$$ H(z) = \frac{z+d}{(z+a)(z+b)} = \frac{d_1}{z+a}+\frac{d_2}{z+b}.$$

To do so, you multiply them by zeros which will cancel one of the poles, and evaluate them at \$z\$ value of the cancelled pole, to obtain those coefficients \$d_1, d_2\$. 
$$ \left[(z+a)H(z)\right]\big\rvert_{z=-a} = d_1,$$
$$ \left[(z+b)H(z)\right]\big\rvert_{z=-b} = d_2,$$
That way, you can find a partial fraction expansion of \$H(z)\$
$$ H(z) = \frac{d_1}{z+a}+\frac{d_2}{z+b}.$$

And since those terms are readily available in a Z transform table, you can convert them to obtain the time-domain of \$H(z)\$
$$ h(n) = \mathcal{Z}^{-1}(\frac{d_1}{z+a})+\mathcal{Z}^{-1}(\frac{d_2}{z+b}).$$