The problem presented involves two congruent rectangles, and the perimeter of the exterior of the figure is given by the equation $P = 4w + 2h$, where $w$ represents the width of each rectangle and $h$ represents the height. The question asks to solve this perimeter equation for $w$.

Given:

$P = 4w + 2h$ We are to solve for $w$.

Steps:

1. Subtract $2h$ from both sides: $P - 2h = 4w$
2. Divide both sides by 4: $w = \frac{P - 2h}{4}$

Thus, the correct expression for $w$ is $w = \frac{P - 2h}{4}$.

This matches the last option in the multiple-choice answers.

Correct answer:

$\boxed{w = \frac{P - 2h}{4}}$

Would you like further clarification or details?

Here are some follow-up questions to deepen understanding:

1. How is the formula for the perimeter of the rectangle derived?
2. What would happen if the two rectangles were not congruent?
3. Can you derive a general formula for the area of this figure using $w$ and $h$?
4. How does the perimeter change if the heights of the rectangles are doubled?
5. Can this concept be extended to three dimensions (e.g., surface area of a 3D shape)?

Tip: When solving for a variable, remember to isolate it step by step using inverse operations, such as addition, subtraction, multiplication, and division.