To find the slope of a line perpendicular to the given line, we first need to determine the slope of the line represented by the equation \( 5x - 2y = -10 \).
We can rewrite this equation in slope-intercept form, \( y = mx + b \), where \( m \) is the slope.
Starting with the equation:
\[ 5x - 2y = -10 \]
We isolate \( y \):
\[ -2y = -5x - 10 \]
Now, divide every term by \(-2\):
\[ y = \frac{5}{2}x + 5 \]
From this form, we can see that the slope \( m \) of the line is:
\[ m = \frac{5}{2} \]
The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, we take the negative reciprocal of \( \frac{5}{2} \):
The negative reciprocal is calculated as follows:
\[ \text{slope of perpendicular line} = -\frac{1}{\frac{5}{2}} = -\frac{2}{5} \]
Thus, the slope of a line perpendicular to the given line is:
\[ \boxed{-\frac{2}{5}} \]