$\displaystyle \displaystyle \begin{align*}a(z - b) &= \frac{cdf}{f + z} - cg - \frac{efh}{f +z } - hi \\ az - ab &= \frac{cdf}{f + z} - \frac{cg(f + z)}{f + z} - \frac{efh}{f + z} - \frac{hi(f + z)}{f + z} \\ az - ab &= \frac{cdf - cg(f + z) - efh - hi(f + z)}{f + z} \\ (az - ab)(f + z) &= cdf - cg(f + z) - efh - hi(f + z) \\ afz + az^2 - abf - abz &= cdf - cgf - cgz - efh - fhi - hiz \\ az^2 + afz + abz + cgz + hiz - abf - cdf + cgf + efh + fhi &= 0 \\ az^2 + \left(af + ab + cg + hi\right)z - abf - cdf + cgf + efh + fhi &= 0 \end{align*}$

This is a quadratic of the form $\displaystyle \displaystyle \alpha z^2 + \beta z + \gamma = 0$, which has solution $\displaystyle \displaystyle z = \frac{-\beta \pm \sqrt{\beta ^2 - 4\alpha\gamma}}{2\alpha}$.

Substitute these values and you finally have your answer.