https://file.aiquickdraw.com/mathbot/file/1726718787375iyvw1se6.jpg

The problem in the image states:

"A rectangle has a perimeter of \( 8x^2 + 10x - 6 \) and has a width of \( 3x^2 + 4x \). What is the length of the rectangle?"

### Solution:

The formula for the perimeter of a rectangle is:
\[
P = 2L + 2W
\]
where:
- \(P\) is the perimeter,
- \(L\) is the length,
- \(W\) is the width.

Given:
\[
P = 8x^2 + 10x - 6
\]
and
\[
W = 3x^2 + 4x.
\]

Substitute these into the perimeter formula:
\[
8x^2 + 10x - 6 = 2L + 2(3x^2 + 4x).
\]
Simplify the right side:
\[
8x^2 + 10x - 6 = 2L + 6x^2 + 8x.
\]
Now, isolate \(2L\):
\[
2L = (8x^2 + 10x - 6) - (6x^2 + 8x),
\]
\[
2L = 8x^2 + 10x - 6 - 6x^2 - 8x,
\]
\[
2L = (8x^2 - 6x^2) + (10x - 8x) - 6,
\]
\[
2L = 2x^2 + 2x - 6.
\]
Now, divide everything by 2 to find \(L\):
\[
L = x^2 + x - 3.
\]

Thus, the length of the rectangle is:
\[
L = x^2 + x - 3.
\]

Would you like more details or have any questions?

### Relative Questions:
1. How would the equation change if the width were \(4x^2 + 5x\)?
2. How can you solve for \(x\) if the perimeter equals a specific value?
3. What happens to the length if \(x = 2\)?
4. How would the solution change if the formula for the perimeter was for a square?
5. What if the width was expressed as a linear function instead of a quadratic?

### Tip:
Always simplify both sides of an equation before isolating variables to reduce the chances of mistakes!

A rectangle has a perimeter of 8x^2 + 10x - 6 and has a width of 3x^2 + 4x. What is the length of the rectangle?

Learn how to find the length of a rectangle with a perimeter of 8x^2 + 10x - 6 and a width of 3x^2 + 4x using algebraic methods. This problem involves simplifying quadratic expressions and applying the perimeter formula, suitable for students in Grades 9-11.

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