Problem

2. The complex numbers $z_1$, $z_2$ and $z_3$ are given by

$z_1 = 2 - i, \quad z_2 = p - i, \quad z_3 = p + i$

where $p$ is a real number.

(a) Find $\dfrac{z_2 z_3}{z_1}$ in the form $a + bi$ where $a$ and $b$ are real. Give your answer in its simplest form in terms of $p$.

Given that $\dfrac{z_2 z_3}{z_1} = 2\sqrt{5}$

(b) find the possible values of $p$.