The image presents a rectangle with the following side lengths:

• One side is represented by $x + 9y$ units.
• The other side is represented by $8x^2 - x + 4y$ units.

To find the perimeter of a rectangle, use the formula:

$\text{Perimeter} = 2 \times (\text{Length} + \text{Width})$

In this case, the length is $8x^2 - x + 4y$ and the width is $x + 9y$.

Now, applying the formula:

$\text{Perimeter} = 2 \times \left( (8x^2 - x + 4y) + (x + 9y) \right)$

Simplifying the expression inside the parentheses:

$\text{Perimeter} = 2 \times \left( 8x^2 - x + x + 4y + 9y \right)$

$\text{Perimeter} = 2 \times \left( 8x^2 + 13y \right)$

Distribute the 2:

$\text{Perimeter} = 16x^2 + 26y$

So, the perimeter of the rectangle is:

$16x^2 + 26y \, \text{units}$

Let me know if you'd like further clarification or if you have additional questions! Here's a set of related questions to explore:

1. What is the area of the rectangle given the same dimensions?
2. How does the perimeter change if the value of $x$ is doubled?
3. Can the perimeter be simplified if $x = 1$ and $y = 2$?
4. How would you calculate the perimeter of a rectangle with one side as $5x + 3y$ and the other as $7x^2 - 4y$?
5. How would the perimeter formula change if the sides were both linear expressions of $x$ and $y$?

Tip: Always combine like terms first when simplifying algebraic expressions to make solving quicker!