We are given a rectangle with a length of $x + 5$ and an area of $2x^3 + 4x^2 - 23x + 35$. We are tasked with finding the width of the rectangle.

The formula for the area of a rectangle is: $\text{Area} = \text{Length} \times \text{Width}$

Given:

• Length $= x + 5$
• Area $= 2x^3 + 4x^2 - 23x + 35$

The width of the rectangle can be found by dividing the area by the length. That is: $\text{Width} = \frac{\text{Area}}{\text{Length}} = \frac{2x^3 + 4x^2 - 23x + 35}{x + 5}$

Now, we will perform polynomial division to divide $2x^3 + 4x^2 - 23x + 35$ by $x + 5$. I will calculate this.The result of dividing the area $2x^3 + 4x^2 - 23x + 35$ by the length $x + 5$ gives a quotient of:

$2x^2 - 6x + 7$

This means the width of the rectangle is $2x^2 - 6x + 7$, which matches one of the answer choices provided.

Do you have any further questions or need additional details?

Here are five related questions you might explore:

1. How does polynomial long division work for more complex polynomials?
2. Can the width expression $2x^2 - 6x + 7$ be factored?
3. How can synthetic division be applied in this case instead of long division?
4. What would happen if the remainder was non-zero during the division?
5. How can we check if the calculated width is correct by multiplying it with the length?

Tip: When dividing polynomials, always check the remainder to verify if it's zero; this ensures the division was exact!