We know the area of the rectangular carpet is given by the expression $10x^2 - 3x - 27$, and the length is $5x - 9$. To find the width of the rectangle, we can use the formula for the area of a rectangle:

$\text{Area} = \text{Length} \times \text{Width}$

Let $w$ be the width. Therefore, we have:

$10x^2 - 3x - 27 = (5x - 9) \times w$

To solve for $w$, we divide both sides of the equation by $5x - 9$:

$w = \frac{10x^2 - 3x - 27}{5x - 9}$

Now, we will perform polynomial division to simplify the expression for the width.

Step 1: Polynomial division of $10x^2 - 3x - 27$ by $5x - 9$

1. Divide the first term: $\frac{10x^2}{5x} = 2x$
2. Multiply $2x$ by $5x - 9$ to get $10x^2 - 18x$
3. Subtract $(10x^2 - 18x)$ from $10x^2 - 3x - 27$, which leaves us with: $(10x^2 - 3x - 27) - (10x^2 - 18x) = 15x - 27$
4. Divide the first term: $\frac{15x}{5x} = 3$
5. Multiply $3$ by $5x - 9$ to get $15x - 27$
6. Subtract $(15x - 27)$ from $15x - 27$, and the remainder is 0.

Thus, the width of the rectangle is:

$w = 2x + 3$

Final Answer:

The width of the rectangle is $2x + 3$ yards.

Would you like more details or have any questions?

Related questions:

1. How do you factor a quadratic expression like $10x^2 - 3x - 27$?
2. What is the process of polynomial division, and how is it used?
3. Can the area formula for a rectangle be applied to other shapes with known dimensions?
4. How can you check if a division of polynomials was performed correctly?
5. How does the quadratic nature of the area expression affect the problem's complexity?

Tip: When performing polynomial division, carefully align terms of the same degree to avoid errors in subtraction and simplification!