Answer
$78$
Work Step by Step
We are given the determinant:
$D=\begin{vmatrix}3&-1&1&2\\-2&0&0&0\\2&-1&-2&3\\1&4&2&3\end{vmatrix}$
In order to compute the determinant we expand it along the second row because it has 3 zeros:
$D=-(-2)D_{21}+0D_{12}-0D_{13}+0D_{14}=2D_{21}$
$=2\begin{vmatrix}-1&1&2\\-1&-2&3\\4&2&3\end{vmatrix}$
$=2\left(-1\begin{vmatrix}-2&3\\2&3\end{vmatrix}-1\begin{vmatrix}-1&3\\4&3\end{vmatrix}+2\begin{vmatrix}-1&-2\\4&2\end{vmatrix}\right)$
$=2[-1(-6-6)-1(-3-12)+2(-2+8)]=2(39)=78$
