$\begin{equation*} \begin{cases} x^{2}_{1} - x^{2}_{2} = 12 &\\ x_1 + x_2 = -2 &\\ x_1x_2 = q & \end{cases} \end{equation*}$
$x^{2}_{1} - x^{2}_{2} = (x_1 - x_2)(x_1 + x_2)$
$x_1 - x_2 = \frac{x^{2}_{1} - x^{2}_{2}}{x_1 + x_1} = \frac{12}{-2} = -6$
$\begin{equation*} \begin{cases} x_1 - x_2 = -6 &\\ x_1 + x_2 = -2 &\\ x_1x_2 = q & \end{cases} \end{equation*}$
$x_1 - x_2 + x_1 + x_2 = -6 - 2$
$2x_1 = -8$
$x_1 = -4$
$-4 + x_2 = -2$
$x_2 = 2$
q = −4 * 2 = −8
$\begin{equation*} \begin{cases} x_1 = -4 &\\ x_2 = 2 &\\ q = -8 & \end{cases} \end{equation*}$