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.