https://file.aiquickdraw.com/mathbot/file/17254505754697x4ab2ww.jpg

The problem shown in the image is about a rectangle. I'll transcribe the problem and solve it step-by-step:

### Given Information:
1. The length of the rectangle is labeled as \( 2x \).
2. The width of the rectangle is labeled as \( x \).

### Part (i) - Expression for Perimeter:
The perimeter of a rectangle is given by the formula:
\[
\text{Perimeter} = 2(\text{length} + \text{width})
\]
Substituting the given values:
\[
\text{Perimeter} = 2(2x + x) = 2(3x) = 6x
\]
Thus, the perimeter of the rectangle is \( 6x \).

### Part (ii) - Show that \( x^2 - 4x - 5 = 0 \):
The task in this part seems to involve solving for \( x \) using some additional information that relates to the equation \( x^2 - 4x - 5 = 0 \). Based on the image, there is likely some connection between the rectangle’s dimensions and this equation. However, the full details of the problem statement in part (ii) are not entirely visible.

To solve the quadratic equation \( x^2 - 4x - 5 = 0 \), we can factor it:

We need two numbers that multiply to \( -5 \) and add to \( -4 \). These numbers are \( -5 \) and \( 1 \). Thus, we factor the equation as:
\[
(x - 5)(x + 1) = 0
\]
Solving for \( x \):
\[
x - 5 = 0 \quad \text{or} \quad x + 1 = 0
\]
\[
x = 5 \quad \text{or} \quad x = -1
\]

Since \( x \) represents a dimension of a rectangle, we discard the negative solution. Therefore, \( x = 5 \).

### Conclusion:
The perimeter of the rectangle is \( 6x \). When \( x = 5 \), the perimeter is \( 6(5) = 30 \).

If you have further questions, feel free to ask! Would you like a detailed explanation on any part of this problem?

Here are some related questions:
1. How do we derive the formula for the perimeter of any polygon?
2. How do you solve more complex quadratic equations using the quadratic formula?
3. What are other methods for solving quadratic equations besides factoring?
4. How do the dimensions of a shape affect its area compared to its perimeter?
5. How can you verify whether the solutions to a quadratic equation are correct?

**Tip**: When solving quadratic equations by factoring, always verify your factors by expanding the factored form back to the original equation.

The perimeter of the rectangle is 6x. The dimensions of the rectangle are labeled as 2x and x. (i) Write an expression for the perimeter in terms of x. (ii) Show that x^2 - 4x - 5 = 0.

Learn how to solve a problem involving the perimeter of a rectangle and a quadratic equation x^2 - 4x - 5. This problem covers essential concepts in geometry and algebra, suitable for students in Grades 7-9.

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