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The expression provided for the total area of the photo and border in square inches is given as:

\[
(10 + 2x)(8 + 2x)
\]
where \( x \) is the width of the border in inches. When expanded, this gives:

\[
(10 + 2x)(8 + 2x) = 80 + 20x + 16x + 4x^2 = 4x^2 + 36x + 80
\]

### Interpretation of \( 4x^2 + 36x \):
The quantity \( 4x^2 + 36x \) represents the **area contribution due to the border** around the photo.

- \( 4x^2 \) corresponds to the area contribution from the corners of the border, where both the width and the height of the border are involved.
- \( 36x \) represents the area contribution from the sides of the border.

The constant term \( 80 \) represents the area of the photo itself, which is \( 10 \times 8 = 80 \) square inches.

### Correct Answer:
The expression \( 4x^2 + 36x \) represents **the area of the border in square inches**.

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Would you like further clarification on this, or do you have any additional questions?

Here are 5 related questions to explore:

1. How would the expression change if the dimensions of the photo were different?
2. How does changing the width \( x \) affect the total area?
3. Can we calculate the area of the border for a specific \( x \) value?
4. What happens to the area expression if the border width is reduced to zero?
5. How would the perimeter of the entire framed picture be calculated?

**Tip**: When expanding expressions involving two terms, make sure to apply the distributive property (FOIL method) carefully!

What does the quantity 4x^2 + 36x represent in the expression for the total area of the photo and border?

Explore how to interpret the quadratic expression 4x^2 + 36x that represents the area of a border around a photo. Learn key algebraic concepts, formulas, and theorems like the Distributive Property. Suitable for students in Grades 6-8.

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