Explanation

(c)
$\because\overrightarrow{\alpha} \times \overrightarrow{\beta} = \left| \begin{matrix}
\widehat{\dot{\mathbf{i}}} & \widehat{\dot{\mathbf{j}}} & \widehat{\mathbf{k}} \\
2 & 3 & - 1 \\
- 1 & 2 & - 4
\end{matrix} \right| = - 10\widehat{\dot{\mathbf{i}}} + 9\widehat{\dot{\mathbf{j}}} + 7\widehat{\mathbf{k}}$
$\therefore\ \left( \overrightarrow{\alpha} \times \overrightarrow{\beta} \right) \bullet \left( \overrightarrow{\alpha} \times \overrightarrow{\gamma} \right) = \left| \begin{matrix}
- 10 & 9 & 7 \\
2 & 3 & - 1 \\
1 & 1 & 1
\end{matrix} \right|$
$= - 10(3 + 1) - 9(2 + 1) + 7(2 - 3)$
$= - 74$
Alternate
$\left( \overrightarrow{\alpha} \times \overrightarrow{\beta} \right) \bullet \left( \overrightarrow{\alpha} \times \overrightarrow{\gamma} \right) = \left| \begin{matrix}
\overrightarrow{\alpha} \bullet \overrightarrow{\alpha} & \overrightarrow{\alpha} \bullet \overrightarrow{\gamma} \\
\overrightarrow{\beta} \bullet \overrightarrow{\alpha} & \overrightarrow{\beta}.\overrightarrow{\gamma}
\end{matrix} \right|\$
$= \left| \begin{matrix}
14 & 4 \\
8 & - 3
\end{matrix} \right| = - 42 - 32$
$= - 74$