Answer
$-10234$
Work Step by Step
Use the property that the determinant will not change when doing proper row or column operations.
$\begin{array}( \\|A|= \\ \\ \end{array}
\begin{vmatrix} -2 & 3 &-1&7\\4 &6 &-2&3\\7 &7 &0&5 \\3 &-12 &4&0\end{vmatrix} $
Column operation: $C_2+3C_3\to C_2$
$\begin{array}( \\|A|= \\ \\ \end{array}
\begin{vmatrix} -2 & 0 &-1&7\\4 &0 &-2&3\\7 &7 &0&5 \\3 &0 &4&0\end{vmatrix} \begin{array}( \\=(-7) \\ \\ \end{array}\begin{vmatrix} -2 &-1&7\\4 &-2&3\\3&4&0\end{vmatrix}$
$|A|=(-7)[3(-3+14)-4(-6-28)]=(-7)[3(13)+4(34)]=-10234$
