Answer
The determinant is $195$.
Work Step by Step
This is a fourth order determinant, so in order to calculate this determinant, split it into a determinant of order 3. Expand along the second column as follows,
$\begin{align}
& \left| \begin{matrix}
-2 & -3 & 3 & 5 \\
1 & -4 & 0 & 0 \\
1 & 2 & 2 & -3 \\
2 & 0 & 1 & 1 \\
\end{matrix} \right|={{\left( -1 \right)}^{1+2}}\left( -3 \right)\left| \begin{matrix}
1 & 0 & 0 \\
1 & 2 & -3 \\
2 & 1 & 1 \\
\end{matrix} \right|+{{\left( -1 \right)}^{2+2}}\left( -4 \right)\left| \begin{matrix}
-2 & 3 & 5 \\
1 & 2 & -3 \\
2 & 1 & 1 \\
\end{matrix} \right|+{{\left( -1 \right)}^{3+2}}2\left| \begin{matrix}
-2 & 3 & 5 \\
1 & 0 & 0 \\
2 & 1 & 1 \\
\end{matrix} \right| \\
& ={{\left( -1 \right)}^{3}}\left( -3 \right)\left| \begin{matrix}
1 & 0 & 0 \\
1 & 2 & -3 \\
2 & 1 & 1 \\
\end{matrix} \right|+{{\left( -1 \right)}^{4}}\left( -4 \right)\left| \begin{matrix}
-2 & 3 & 5 \\
1 & 2 & -3 \\
2 & 1 & 1 \\
\end{matrix} \right|+{{\left( -1 \right)}^{5}}2\left| \begin{matrix}
-2 & 3 & 5 \\
1 & 0 & 0 \\
2 & 1 & 1 \\
\end{matrix} \right| \\
& =\left( -1 \right)\left( -3 \right)\left| \begin{matrix}
1 & 0 & 0 \\
1 & 2 & -3 \\
2 & 1 & 1 \\
\end{matrix} \right|+\left( 1 \right)\left( -4 \right)\left| \begin{matrix}
-2 & 3 & 5 \\
1 & 2 & -3 \\
2 & 1 & 1 \\
\end{matrix} \right|+\left( -1 \right)2\left| \begin{matrix}
-2 & 3 & 5 \\
1 & 0 & 0 \\
2 & 1 & 1 \\
\end{matrix} \right| \\
& =3\left| \begin{matrix}
1 & 0 & 0 \\
1 & 2 & -3 \\
2 & 1 & 1 \\
\end{matrix} \right|+\left( -4 \right)\left| \begin{matrix}
-2 & 3 & 5 \\
1 & 2 & -3 \\
2 & 1 & 1 \\
\end{matrix} \right|+\left( -2 \right)\left| \begin{matrix}
-2 & 3 & 5 \\
1 & 0 & 0 \\
2 & 1 & 1 \\
\end{matrix} \right|
\end{align}$
Now calculate each third order determinant as follows:
First consider,
$\begin{align}
& 3\left| \begin{matrix}
1 & 0 & 0 \\
1 & 2 & -3 \\
2 & 1 & 1 \\
\end{matrix} \right|=3\left( 1\left| \begin{matrix}
2 & -3 \\
1 & 1 \\
\end{matrix} \right|-1\left| \begin{matrix}
0 & 0 \\
1 & 1 \\
\end{matrix} \right|+2\left| \begin{matrix}
0 & 0 \\
2 & -3 \\
\end{matrix} \right| \right) \\
& =3\left( 1\left[ 2\left( 1 \right)-1\left( -3 \right) \right]-1\left[ 0\left( 1 \right)-1\left( 0 \right) \right]+2\left[ 0\left( -3 \right)-2\left( 0 \right) \right] \right) \\
& =3\left( 1\left[ 2+3 \right]-1\left( 0 \right)+2\left( 0 \right) \right) \\
& =15
\end{align}$
Next consider the second determinant,
$\begin{align}
& \left( -4 \right)\left| \begin{matrix}
-2 & 3 & 5 \\
1 & 2 & -3 \\
2 & 1 & 1 \\
\end{matrix} \right|=\left( -4 \right)\left( \left( -2 \right)\left| \begin{matrix}
2 & -3 \\
1 & 1 \\
\end{matrix} \right|-1\left| \begin{matrix}
3 & 5 \\
1 & 1 \\
\end{matrix} \right|+2\left| \begin{matrix}
3 & 5 \\
2 & -3 \\
\end{matrix} \right| \right) \\
& =\left( -4 \right)\left( \left( -2 \right)\left[ 2\left( 1 \right)-1\left( -3 \right) \right]-1\left[ 3\left( 1 \right)-1\left( 5 \right) \right]+2\left[ 3\left( -3 \right)-2\left( 5 \right) \right] \right) \\
& =\left( -4 \right)\left( -2\left[ 2+3 \right]-1\left[ 3-5 \right]+2\left[ -9-10 \right] \right) \\
& =\left( -4 \right)\left( -10+2-38 \right)
\end{align}$
Simplify this to get,
$\begin{align}
& \left( -4 \right)\left| \begin{matrix}
-2 & 3 & 5 \\
1 & 2 & -3 \\
2 & 1 & 1 \\
\end{matrix} \right|=\left( -4 \right)\left( -10+2-38 \right) \\
& =\left( -4 \right)\left( -46 \right) \\
& =184
\end{align}$
At last consider the third determinant,
$\begin{align}
& \left( -2 \right)\left| \begin{matrix}
-2 & 3 & 5 \\
1 & 0 & 0 \\
2 & 1 & 1 \\
\end{matrix} \right|=\left( -2 \right)\left( \left( -2 \right)\left| \begin{matrix}
0 & 0 \\
1 & 1 \\
\end{matrix} \right|-1\left| \begin{matrix}
3 & 5 \\
1 & 1 \\
\end{matrix} \right|+2\left| \begin{matrix}
3 & 5 \\
0 & 0 \\
\end{matrix} \right| \right) \\
& =\left( -2 \right)\left( \left( -2 \right)\left[ 0\left( 1 \right)-1\left( 0 \right) \right]-1\left[ 3\left( 1 \right)-1\left( 5 \right) \right]+2\left[ 3\left( 0 \right)-0\left( 5 \right) \right] \right) \\
& =\left( -2 \right)\left( \left( -2 \right)\left( 0 \right)-1\left( 3-5 \right)+2\left( 0 \right) \right) \\
& =-4
\end{align}$
Now put these values in the original equations and simplify:
$\begin{align}
& \left| \begin{matrix}
-2 & -3 & 3 & 5 \\
1 & -4 & 0 & 0 \\
1 & 2 & 2 & -3 \\
2 & 0 & 1 & 1 \\
\end{matrix} \right|=3\left| \begin{matrix}
1 & 0 & 0 \\
1 & 2 & -3 \\
2 & 1 & 1 \\
\end{matrix} \right|+\left( -4 \right)\left| \begin{matrix}
-2 & 3 & 5 \\
1 & 2 & -3 \\
2 & 1 & 1 \\
\end{matrix} \right|+\left( -2 \right)\left| \begin{matrix}
-2 & 3 & 5 \\
1 & 0 & 0 \\
2 & 1 & 1 \\
\end{matrix} \right| \\
& =15+184-4 \\
& =195
\end{align}$