We are tasked with finding the coordinates of $Q$ after a reflection across the vertical line $x = -3$.

Steps to Solve:

1. Understanding Reflection:

   • When a point is reflected across a vertical line $x = k$, the x-coordinate changes symmetrically about $x = k$.
   • The y-coordinate stays the same because the reflection happens horizontally.

2. Given Points:

   • $P'(5, 2)$ and $Q'(2, 4)$ are the reflected images of $P$ and $Q$, respectively.
   • We need to find $Q$'s original coordinates before reflection.

3. Finding $Q$'s Coordinates: To reflect a point back across $x = -3$, follow the symmetry rule:

   • Let $x_1$ be the original x-coordinate of $Q$ and $x_2$ be the reflected coordinate. The midpoint between them is $-3$, the line of reflection: $\frac{x_1 + x_2}{2} = -3$
   • Substituting $x_2 = 2$ (from $Q'(2, 4)$): $\frac{x_1 + 2}{2} = -3$
   • Multiply through by 2: $x_1 + 2 = -6$
   • Solve for $x_1$: $x_1 = -8$

   The y-coordinate remains the same as in $Q'$, which is $4$.

4. Final Answer: The original coordinates of $Q$ are: $(-8, 4)$ Therefore, the correct choice is D (-8, 4).

Do you need a detailed explanation for the reflection process? Let me know!

Related Questions:

1. How do you reflect a point across any vertical line $x = k$?
2. What happens to coordinates when reflecting across a horizontal line $y = k$?
3. How can you determine the line of reflection if both original and reflected points are known?
4. How do you reflect a point across the y-axis or x-axis?
5. What is the geometric significance of symmetry in reflections?

Tip: Reflections maintain equal distances from the line of reflection while preserving the y-coordinate (for vertical reflections).