Answer
$-173$
Work Step by Step
We are given the determinant:
$D=\begin{vmatrix}-2&-3&3&5\\1&-4&0&0\\1&2&2&-3\\2&0&1&1\end{vmatrix}$
In order to compute the determinant we expand it along the second row because it has 2 zeros:
$D=-1D_{21}+(-4)D_{12}-0D_{13}+0D_{14}=-D_{21}4D_{22}$
$=-\begin{vmatrix}-3&3&5\\2&2&-3\\0&1&1\end{vmatrix}-4\begin{vmatrix}-2&3&5\\1&2&-3\\2&1&1\end{vmatrix}$
$=-\left(-0\begin{vmatrix}3&5\\2&-3\end{vmatrix}-1\begin{vmatrix}-3&5\\2&-3\end{vmatrix}+1\begin{vmatrix}-3&3\\2&2\end{vmatrix}\right)-4\left(-2\begin{vmatrix}2&-3\\1&1\end{vmatrix}-3\begin{vmatrix}1&-3\\2&1\end{vmatrix}+5\begin{vmatrix}1&2\\2&1\end{vmatrix}\right)$
$=-[0-1(9-10)+1(-6-6)]-4[-2(2+3)-3(1+6)+5(1-4)]$
$=11-184=-173$
