First, we start with the equation of line \( g \):
\[ y - 8 = -2(x - 4) \]
To put this into slope-intercept form (y = mx + b), we can simplify it:
- Distribute \(-2\) on the right side:
\[ y - 8 = -2x + 8 \]
- Add \(8\) to both sides:
\[ y = -2x + 16 \]
The slope (\(m\)) of line \(g\) is \(-2\). Since line \(h\) is parallel to line \(g\), it will have the same slope of \(-2\).
Now, we need to find the equation of line \(h\) that passes through the point \((-5, 1)\). We can use the point-slope form of a line equation, which is:
\[ y - y_1 = m(x - x_1) \]
Substituting in \(m = -2\) and the point \((-5, 1)\):
\[ y - 1 = -2(x - (-5)) \]
This simplifies to:
\[ y - 1 = -2(x + 5) \]
Distributing \(-2\):
\[ y - 1 = -2x - 10 \]
Adding \(1\) to both sides gives us:
\[ y = -2x - 9 \]
Thus, the equation of line \(h\) in slope-intercept form is:
\[ \boxed{y = -2x - 9} \]