The image shows a mathematical problem asking to find the length of a rectangle, given its area and width.

The problem can be summarized as follows:

• Area of the rectangle: $8a^2 + 18a - 6$
• Width of the rectangle: $4a - 2$
• The task is to find the length by using the formula for the area of a rectangle and applying long division.

Solution:

We know the formula for the area of a rectangle is:

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

Given:

• Area: $8a^2 + 18a - 6$
• Width: $4a - 2$

To find the length, we divide the area by the width using polynomial long division:

$\text{Length} = \frac{8a^2 + 18a - 6}{4a - 2}$

I'll now perform the long division to find the length.

Long Division Steps:

1. Divide the leading term of the numerator $8a^2$ by the leading term of the denominator $4a$, which gives $2a$.

2. Multiply $2a$ by the entire divisor $4a - 2$, which results in $8a^2 - 4a$.

3. Subtract $(8a^2 - 4a)$ from $(8a^2 + 18a - 6)$, leaving $22a - 6$.

4. Divide the new leading term $22a$ by $4a$, which gives $+5.5$ (or $+\frac{11}{2}$).

5. Multiply $5.5$ by $4a - 2$, resulting in $22a - 11$.

6. Subtract $(22a - 11)$ from $(22a - 6)$, resulting in $5$.

Thus, the quotient (length) is $2a + 5.5$, with a remainder of 5.

This gives:

$\text{Length} = 2a + \frac{11}{2} \quad \text{with a remainder of 5.}$

This represents the length of the rectangle based on the division process.

Do you want more details on the long division steps, or do you have any other questions?

Related Questions:

1. How is polynomial long division different from regular long division?
2. Can the remainder in polynomial division always be interpreted geometrically?
3. How do you simplify polynomials before dividing them?
4. What other methods, besides long division, can be used to divide polynomials?
5. How can we check the result of polynomial division?

Tip:

Always check your result by multiplying the quotient and the divisor back together to ensure they produce the original dividend.