Let's solve this step-by-step.

1. Initial Point: The initial point is $(5, -1)$.

2. Translation:

   • Moving 5 units to the right adds 5 to the $x$-coordinate: $5 + 5 = 10$.
   • Moving 3 units down subtracts 3 from the $y$-coordinate: $-1 - 3 = -4$.
   • The new point after translation is $(10, -4)$.

3. Reflection:

   • Reflecting across the line $x = 0$ (the $y$-axis) changes the sign of the $x$-coordinate but leaves the $y$-coordinate the same.
   • So, the reflection of $(10, -4)$ across $x = 0$ is $(-10, -4)$.

4. Quadrant Identification:

   • The point $(-10, -4)$ lies in Quadrant III (where both $x$ and $y$ are negative).

Answer: The correct answer is C. Quadrant III.

Would you like more details on any of the steps?

Here are 5 related questions to expand your understanding:

1. What would happen if the reflection was applied across the $y$-axis instead?
2. How would you translate a point by a different combination of units?
3. How can we reflect a point across the line $y = x$?
4. What is the effect of reflecting a point across both the $x$-axis and $y$-axis?
5. How can we determine the quadrant of a point based on its coordinates?

Tip: Remember that reflecting across the line $x = 0$ simply changes the sign of the $x$-coordinate, while reflecting across $y = 0$ changes the sign of the $y$-coordinate.