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!