To find the perimeter of the rectangle, we need to use the formula for the perimeter of a rectangle:

$P = 2(\text{length} + \text{width})$

Step 1: Identify the length and width

From the problem:

• Length: $6x^2 - x + 7y$ units
• Width: $x + 8y$ units

Step 2: Plug into the formula

$P = 2 \left( (6x^2 - x + 7y) + (x + 8y) \right)$

Step 3: Simplify the terms inside the parentheses

Combine like terms: $(6x^2 - x + 7y) + (x + 8y) = 6x^2 - x + x + 7y + 8y = 6x^2 + 15y$

Step 4: Multiply by 2

$P = 2(6x^2 + 15y) = 12x^2 + 30y$

Final Answer:

The perimeter is: $\boxed{12x^2 + 30y \, \text{units}}$

Would you like a further breakdown of the steps or have any questions?

Related Questions:

1. What would the perimeter be if $x = 2$ and $y = 3$?
2. How do you find the area of this rectangle?
3. What is the significance of combining like terms when simplifying expressions?
4. Can a rectangle have negative dimensions in algebra?
5. How do we extend this concept to find the perimeter of other shapes?

Tip:

Always double-check your work by substituting values to verify the accuracy of the final expression.