The co-factor of the element '4' in the determinant
$$\left|\begin{array}{cccc}1& 3& 5& 1\\ 2& 3& 4& 2\\ 8& 0& 1& 1\\ 0& 2& 1& 1\end{array}\right|$$

The element '4' is at position $a_{23}$ (2nd row, 3rd column).The cofactor $C_{23}=(-1{)}^{2+3}{M}_{23}=-M_{23}$.$M_{23}$ is the determinant of the matrix formed by deleting the 2nd row and 3rd column:
$$M_{23}=\left|\begin{array}{ccc}1& 3& 1\\ 8& 0& 1\\ 0& 2& 1\end{array}\right|$$
Expanding along the first row:
$$M_{23}=1(0-2)-3(8-0)+1(16-0)$$
$$M_{23}=-2-24+16=-10$$
Therefore, $C_{23}=-(-10)=10$.