Let's break down the problem step by step:

We have a square and a rectangle with the following information:

1. The width of the rectangle is the same as the side length of the square, both denoted as $x$.
2. The length of the rectangle is one foot more than twice its width, which makes the length of the rectangle $2x + 1$.
3. The perimeter of the rectangle is 26 feet more than the perimeter of the square.

Part (a): Write an expression for the length of the rectangle

The length of the rectangle is described as "one foot more than twice its width." Since the width of the rectangle is $x$, the length is: $\text{Length of rectangle} = 2x + 1$ This expression matches the label in the drawing.

Part (b): Show that 5 could not be the value of $x$

We need to substitute $x = 5$ into the problem and show that it does not work.

1. Perimeter of the square: The perimeter of a square with side length $x$ is $4x$. So if $x = 5$, the perimeter of the square is: $4(5) = 20 \text{ feet}$

2. Perimeter of the rectangle: The perimeter of a rectangle is given by $2 \times (\text{length} + \text{width})$. The width is $x = 5$, and the length is $2(5) + 1 = 11$. Therefore, the perimeter of the rectangle is: $2(5 + 11) = 2(16) = 32 \text{ feet}$

3. Condition: The problem states that the perimeter of the rectangle is 26 feet more than that of the square. If the perimeter of the square is 20 feet, then the perimeter of the rectangle should be: $20 + 26 = 46 \text{ feet}$ However, we found that the perimeter of the rectangle is only 32 feet. Since this does not match the condition, $x = 5$ cannot be the value of $x$.

Part (c): Set up an equation and solve it to find the value of $x$

We need to set up an equation based on the perimeter condition.

1. Perimeter of the square: The perimeter is $4x$.
2. Perimeter of the rectangle: The perimeter is $2(x + (2x + 1)) = 2(3x + 1) = 6x + 2$.
3. Condition: The perimeter of the rectangle is 26 feet more than that of the square. So we can set up the equation: $6x + 2 = 4x + 26$
4. Solve the equation: $6x + 2 = 4x + 26$ Subtract $4x$ from both sides: $2x + 2 = 26$ Subtract 2 from both sides: $2x = 24$ Divide by 2: $x = 12$

Thus, the value of $x$ is 12.

Would you like further clarification on any part of the solution? Here are 5 related questions:

1. How do we find the perimeter of a rectangle?
2. What other shapes have similar relationships between perimeter and dimensions?
3. How would the problem change if the rectangle's length were three times the width instead of twice the width?
4. Could $x$ be a negative number in this context?
5. What is the relationship between the perimeter and area of the square and rectangle?

Tip: When solving word problems, try to identify and organize the known and unknown values into equations based on given relationships. This helps break down the problem more easily.