To solve this problem, let’s go through the details provided in the text and diagram.

Problem Analysis:

1. Structure: We have a rectangular prism with points labeled, where point $P$ lies on the extended edge of $AE$.
2. Line $CP$: It intersects the face diagonal $EG$ at point $S$.
3. Given: $ES = 2$.
4. Objective: Find the distance $EP$.

Solution Outline:

1. Establish Coordinates and Dimensions:

   • From the image, we have dimensions along the edges: $AB = 3$, $AD = 4$, and $AE = 5$.
   • Assign coordinates for clarity:
     • $A = (0, 0, 0)$
     • $B = (3, 0, 0)$
     • $D = (0, 4, 0)$
     • $E = (0, 0, 5)$
     • $G = (3, 4, 5)$

2. Calculate Vector for $EG$:

   • Vector $EG = G - E = (3, 4, 0)$.

3. Point $S$:

   • Since $S$ divides $EG$ with $ES = 2$, we can set up a ratio for $S$'s position on $EG$.
   • Use the section formula to find $S$'s coordinates.

4. Distance $EP$:

   • Use the distance formula to find the length of $EP$ once coordinates for $P$ are confirmed.

I'll proceed with calculations based on these steps to determine $EP$.The distance $EP$ is calculated to be $5$ units.

Would you like further details or have any other questions?

Here are five related questions to explore further:

1. How would the calculation change if $ES$ had a different length?
2. Can we generalize this method for any point on an extended line in 3D space?
3. What if $P$ were on a different edge of the rectangular prism?
4. How would you calculate distances for points lying on a diagonal within the prism?
5. How can vector methods simplify the calculations in geometry problems like these?

Tip: Using vectors and the section formula can greatly simplify finding points that divide a line segment in a given ratio in 3D geometry.