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Let's break down and solve the questions in the case study provided:

### 1. What is the length and width of the garden when \( x = 3 \)?

The length of the garden is given by \( x + 2 \), and the width by \( x - 1 \).
Substituting \( x = 3 \):

\[
\text{Length} = 3 + 2 = 5 \text{ meters}
\]
\[
\text{Width} = 3 - 1 = 2 \text{ meters}
\]

Thus, the length is 5 meters and the width is 2 meters.

### 2. Verify whether \( A(3) = 14 \) is correct by calculating the area using the given polynomial.

The area polynomial is \( A(x) = x^2 + x - 2 \).

Substituting \( x = 3 \) into the polynomial:

\[
A(3) = 3^2 + 3 - 2 = 9 + 3 - 2 = 10
\]

Thus, \( A(3) = 10 \), not 14. So, \( A(3) = 14 \) is incorrect.

### 3. Factorize the polynomial \( A(x) = x^2 + x - 2 \).

We need to factor \( x^2 + x - 2 \). To do this, we look for two numbers whose product is \(-2\) (the constant term) and whose sum is \(1\) (the coefficient of \(x\)).

The correct factorization is:

\[
A(x) = (x + 2)(x - 1)
\]

### 4. Find the dimensions of the garden if the area is 0.

If the area is 0, we set the polynomial equal to 0:

\[
A(x) = x^2 + x - 2 = 0
\]
From the factorization, we have:

\[
(x + 2)(x - 1) = 0
\]

Setting each factor equal to 0:

\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]
\[
x - 1 = 0 \quad \Rightarrow \quad x = 1
\]

Thus, the values of \( x \) for which the area is 0 are \( x = -2 \) and \( x = 1 \).

For these values:
- When \( x = -2 \), the length is \( -2 + 2 = 0 \) and the width is \( -2 - 1 = -3 \), so the garden has no length.
- When \( x = 1 \), the length is \( 1 + 2 = 3 \) and the width is \( 1 - 1 = 0 \), so the garden has no width.

Would you like further details or explanations?

### Related Questions:
1. What is the area of the garden when \( x = 4 \)?
2. How would the polynomial change if the width were \( x + 1 \)?
3. What happens to the area as \( x \) becomes negative?
4. Solve for the value of \( x \) when the area is 20 square meters.
5. How can we graph the area function \( A(x) = x^2 + x - 2 \)?

### Tip:
Always check the sign of \( x \) when solving real-world problems like dimensions. Negative values for length or width usually don't make sense physically.

Learn how to solve polynomial area problems using quadratic equations and factoring. This case study involves calculating the area of a rectangular garden, factoring polynomials, and finding dimensions based on a given area function. Suitable for students in Grades 9-10.

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