To find the equation of a line that is parallel to \( y = -4x - 5 \) and passes through the point \((-2, 6)\), we need to follow these steps:
1. Identify the slope of the given line:
The equation of the given line is \( y = -4x - 5 \). This equation is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope. For the given line, the slope (\( m \)) is \(-4\).
2. Use the slope for the new line:
Since parallel lines have the same slope, the slope of our new line (that we are looking to find) will also be \(-4\).
3. Use the point-slope form of the line equation:
The point-slope form of a line's equation is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where \( (x_1, y_1) \) is a point on the line, and \( m \) is the slope. For our problem, \( (x_1, y_1) = (-2, 6) \) and \( m = -4 \).
4. Plug in the values:
Substitute the point \((-2, 6)\) and the slope \(-4\) into the point-slope form equation:
[tex]\[
y - 6 = -4(x + 2)
\][/tex]
5. Simplify the equation:
Distribute the \(-4\) on the right-hand side:
[tex]\[
y - 6 = -4x - 8
\][/tex]
Add \(6\) to both sides to solve for \( y \):
[tex]\[
y = -4x - 8 + 6
\][/tex]
[tex]\[
y = -4x - 2
\][/tex]
This describes the new line that is parallel to the given line and passes through the point \((-2, 6)\). Therefore, the equation of the line is:
[tex]\[ \boxed{y = -4x - 2} \][/tex]
Hence, the correct option is [tex]\( \text{D.} \ y = -4x - 2 \)[/tex].
