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Let's solve the problem step by step.

Given the rectangle with the following dimensions:
- One side is labeled \(4x + 5\).
- The opposite side is labeled \(4x - 2\).
- Another side is labeled \(x - 4\).

Since opposite sides of a rectangle are equal, we can set up the equation:
\[ 4x - 2 = x - 4 \]

To solve for \( x \), we perform the following steps:

1. Subtract \( x \) from both sides:
\[ 4x - 2 - x = x - 4 - x \]
\[ 3x - 2 = -4 \]

2. Add 2 to both sides:
\[ 3x - 2 + 2 = -4 + 2 \]
\[ 3x = -2 \]

3. Divide both sides by 3:
\[ x = -\frac{2}{3} \]

However, the given answer states that \( x = 6 \). Let's recheck the steps or any given details in the problem for a possible mistake.

Revisiting the problem:

Given dimensions:
\[ 4x - 2 \]
\[ 4x + 5 \]
\[ x - 4 \]

Reevaluating the equation \( 4x - 2 = x - 4 \):

1. Subtract \( x \) from both sides:
\[ 4x - 2 - x = x - 4 - x \]
\[ 3x - 2 = -4 \]

2. Add 2 to both sides:
\[ 3x - 2 + 2 = -4 + 2 \]
\[ 3x = -2 \]

3. Divide both sides by 3:
\[ x = -\frac{2}{3} \]

To verify:
\[ 4x - 2 = x - 4 \]
Substituting \( x = 6 \):
\[ 4(6) - 2 = 6 - 4 \]
\[ 24 - 2 = 2 \]
\[ 22 \neq 2 \]

Since the equation doesn't hold, there must be a mistake in the given answers. Please verify if there is any other information provided or correct the given answers.

Would you like me to attempt another approach or verify any additional details?

Do you have any questions or need further details?

Here are some related questions for further practice:
1. What is the area of the rectangle when \( x = 6 \)?
2. Find the length of each side if \( x = 3 \).
3. Determine the perimeter if \( x = 2 \).
4. What happens to the perimeter if \( x \) is doubled?
5. Find the value of \( x \) if the perimeter is 50.
6. What is the perimeter if \( x = 0 \)?
7. Determine the dimensions of the rectangle if \( x = -1 \).
8. What value of \( x \) makes the area of the rectangle 100 square units?

**Tip:** Always double

Explore how to solve equations involving rectangle dimensions. This problem covers algebraic equations related to geometric shapes, specifically rectangles. Suitable for Grades 7-9 students learning about equations and geometry in mathematics.

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