https://file.aiquickdraw.com/mathbot/file/1738265860653expqy3h1.jpg

The image you uploaded shows a rectangle with expressions for its length and width. We are tasked with finding its perimeter. The formula for the perimeter of a rectangle is:

\[
P = 2 \times (\text{length} + \text{width})
\]

From the image, we have:
- Length = \( x + 10y \)
- Width = \( 4x^2 - x + 5y \)

Now, using the perimeter formula:

\[
P = 2 \times \left( (x + 10y) + (4x^2 - x + 5y) \right)
\]

Let's simplify this expression step-by-step.

1. Combine like terms inside the parentheses:
\[
(x + 10y) + (4x^2 - x + 5y) = 4x^2 + (x - x) + (10y + 5y)
\]
\[
= 4x^2 + 15y
\]

2. Now multiply by 2:
\[
P = 2 \times (4x^2 + 15y)
\]
\[
P = 8x^2 + 30y
\]

Thus, the perimeter is:

\[
\boxed{8x^2 + 30y} \text{ units}
\]

Would you like further explanation or details on any steps? 

Here are some follow-up questions:
1. How does the perimeter change if the expressions for length or width are modified?
2. What is the significance of simplifying expressions before calculating the perimeter?
3. Can you calculate the area of the same rectangle using these expressions?
4. How would the perimeter change if the length was halved?
5. What are the properties of the expressions used for length and width in this problem?

**Tip**: Always simplify algebraic expressions before substituting into formulas for easier calculations.

Find the perimeter of a rectangle with length (x + 10y) and width (4x^2 - x + 5y).

Learn how to find the perimeter of a rectangle given algebraic expressions for its length and width. This problem involves polynomial simplification and the perimeter formula. Suitable for students in Grades 7-10.

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