Hence verify that \(\sqrt { 5 } - \mathrm { i }\) is a root of the equation
$$( 2 + \mathrm { i } \sqrt { 5 } ) z = 3 z ^ { * }$$
where \(z ^ { * }\) is the conjugate of \(z\).
The quadratic equation
$$x ^ { 2 } + p x + q = 0$$
in which the coefficients \(p\) and \(q\) are real, has a complex root \(\sqrt { 5 } - \mathrm { i }\).
Write down the other root of the equation.
Find the sum and product of the two roots of the equation.