Explanation

(b)
$|z| = \left| \left( z - \frac{4}{z} \right) + \frac{4}{z} \right|$
$\Rightarrow \ \ \ \ \ \ |z| \leq \left| z - \frac{4}{z} \right| + \frac{4}{|z|}$
$\Rightarrow \ \ \ \ \ \ |z| \leq 2 + \frac{4}{|z|}$
$\Rightarrow \ \ \ \ \ \ \ \ \left( |z|^{2} - \left( \sqrt{5} + 1 \right) \right)\left( |z| - \left( 1 - \sqrt{5} \right) \right) \leq 0$
$\Rightarrow \ \ \ \ \ 1 - \sqrt{5} \leq |z| \leq \sqrt{5} + 1$