The polynomial \(x^3 + 5x^2 + 31x + 75\) is denoted by \(p(x)\).

(a) Show that \((x + 3)\) is a factor of \(p(x)\).

(b) Show that \(z = -1 + 2\sqrt{6}i\) is a root of \(p(z) = 0\).

(c) Hence find the complex numbers \(z\) which are roots of \(p(z^2) = 0\).