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From the image you uploaded, the problem is asking you to find the width of a rectangle given its length and area. Here's a summary of the key information:

- **Length**: \( 3x + 2 \)
- **Area**: \( 21x^2 - x - 10 \)

To find the width, we use the formula for the area of a rectangle:
\[
\text{Area} = \text{Length} \times \text{Width}
\]
So, the width can be found by dividing the area by the length:
\[
\text{Width} = \frac{\text{Area}}{\text{Length}} = \frac{21x^2 - x - 10}{3x + 2}
\]

Now, let's perform the division and simplify the expression.

### Step 1: Perform Polynomial Division
We need to divide \( 21x^2 - x - 10 \) by \( 3x + 2 \) using polynomial long division.

#### Step 2: Division Process
1. Divide the leading term of the numerator \( 21x^2 \) by the leading term of the denominator \( 3x \), which gives \( 7x \).
2. Multiply \( 7x \) by \( 3x + 2 \), which gives \( 21x^2 + 14x \).
3. Subtract \( (21x^2 + 14x) \) from \( 21x^2 - x - 10 \), which gives \( -15x - 10 \).
4. Divide \( -15x \) by \( 3x \), which gives \( -5 \).
5. Multiply \( -5 \) by \( 3x + 2 \), which gives \( -15x - 10 \).
6. Subtract \( (-15x - 10) \) from \( -15x - 10 \), which gives \( 0 \).

Thus, the result of the division is:
\[
\frac{21x^2 - x - 10}{3x + 2} = 7x - 5
\]

### Final Answer:
The width of the rectangle is \( 7x - 5 \).

Would you like more details or have any other questions about the process?

Here are five related questions to consider:
1. How do you perform polynomial long division?
2. Can you verify the result by multiplying \( (3x + 2) \) and \( (7x - 5) \)?
3. What happens if the area is expressed differently, such as using different powers of \( x \)?
4. How would the process change if the leading term of the divisor had a higher degree?
5. How can factoring be used as an alternative to polynomial division?

**Tip**: When performing polynomial long division, always check your result by multiplying the quotient by the divisor to ensure accuracy.

The length of a rectangle is 3x + 2. Its area is 21x^2 - x - 10. To find an expression that represents the width of the rectangle, perform the following operation and simplify completely: (21x^2 - x - 10) / (3x + 2).

Learn how to find the width of a rectangle when given its length and area by using polynomial division. This problem involves dividing (21x^2 - x - 10) by (3x + 2) and simplifying the result, covering key algebraic concepts suitable for students in Grades 9-11.

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