The problem is asking to find the equation of a line in two forms—point-slope form and slope-intercept form. Here's how to approach this:

Step 1: Understand the slope of the parallel line

The equation of the given line is $y = -2x + 2$. The slope of this line is $m = -2$.

Since the new line is parallel to the given line, it will have the same slope: $m = -2$.

Step 2: Write the equation in point-slope form

The line passes through the point $(-4, -6)$. The point-slope form of a line is: $y - y_1 = m(x - x_1)$ Substitute $m = -2$, $x_1 = -4$, and $y_1 = -6$: $y - (-6) = -2(x - (-4))$ Simplify: $y + 6 = -2(x + 4)$ This is the point-slope form of the line.

Step 3: Convert to slope-intercept form

To convert to slope-intercept form $y = mx + b$, expand and simplify the equation: $y + 6 = -2(x + 4)$ Distribute $-2$: $y + 6 = -2x - 8$ Subtract 6 from both sides: $y = -2x - 14$ This is the slope-intercept form of the line.

Final Answer:

1. Point-slope form: $y + 6 = -2(x + 4)$
2. Slope-intercept form: $y = -2x - 14$

Would you like more details or have additional questions?

Here are 5 related questions to deepen your understanding:

1. How do you find the slope of a line if only two points are given?
2. What changes when a line is perpendicular instead of parallel to a given line?
3. Can you derive the point-slope form equation of a line passing through $(0, 0)$?
4. How do you verify if two lines are parallel or perpendicular?
5. How can slope-intercept form help in graphing a line quickly?

Tip: Always check the slope carefully when dealing with parallel or perpendicular lines—it's key to writing the correct equation!