Using a cofactor/Laplace expansion along the third row:
[tex]\begin{vmatrix}-2&2&-5\\4&-1&-2\\1&3&-5\end{vmatrix}=1\begin{vmatrix}2&-5\\-1&-2\end{vmatrix}-3\begin{vmatrix}-2&-5\\4&-2\end{vmatrix}+(-5)\begin{vmatrix}-2&2\\4&-1\end{vmatrix}[/tex]
The determinant of a generic 2x2 matrix is
[tex]\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc[/tex]
So we get
[tex]\begin{vmatrix}-2&2&-5\\4&-1&-2\\1&3&-5\end{vmatrix}=1(-9)-3(24)+(-5)(-6)=\boxed{-51}[/tex]
