To find the slope of the line given by the equation \( y - 10 = -5 ( x + 20) \), we can either rewrite the equation in slope-intercept form \( y = mx + b \), where \( m \) is the slope, or we can directly identify the slope from the equation.
Starting with the given equation:
\[ y - 10 = -5 (x + 20) \]
First, distribute the -5 on the right-hand side:
\[ y - 10 = -5x - 100 \]
Next, add 10 to both sides to isolate \( y \):
\[ y = -5x - 100 + 10 \]
This simplifies to:
\[ y = -5x - 90 \]
Now, the equation is in slope-intercept form \( y = mx + b \), where \( m \) is the slope. Here, the slope \( m = -5 \).
Thus, the slope of the line is:
\[ \text{C: } -5 \]