https://file.aiquickdraw.com/mathbot/file/1726839727196f9lmz3po.jpg

To solve this problem, we need to find the expression that represents the area of the given rectangle.

The **area** of a rectangle is given by the formula:

\[
\text{Area} = \text{length} \times \text{width}
\]

From the image:
- The **length** is \( 2x + 10 \)
- The **width** is \( 50 - x \)

### Step 1: Multiply the two expressions
We multiply the two expressions for the length and width:

\[
\text{Area} = (2x + 10)(50 - x)
\]

### Step 2: Apply the distributive property
Now, distribute each term in \( 2x + 10 \) to \( 50 - x \):

\[
(2x + 10)(50 - x) = 2x(50 - x) + 10(50 - x)
\]

### Step 3: Simplify each term
First, simplify \( 2x(50 - x) \):

\[
2x(50 - x) = 2x \cdot 50 - 2x \cdot x = 100x - 2x^2
\]

Next, simplify \( 10(50 - x) \):

\[
10(50 - x) = 10 \cdot 50 - 10 \cdot x = 500 - 10x
\]

### Step 4: Combine like terms
Now, combine the terms from the two parts:

\[
100x - 2x^2 + 500 - 10x = -2x^2 + 90x + 500
\]

### Step 5: Identify the correct answer
The expression for the area is \( -2x^2 + 90x + 500 \), which matches option **A**.

Thus, the correct answer is **A**.

---

Would you like further details or have any questions?

Here are five related questions to explore:
1. How would the area expression change if the length was doubled?
2. What happens to the area when \( x = 0 \)?
3. Can you factor the expression \( -2x^2 + 90x + 500 \)?
4. What is the maximum or minimum area value for this rectangle, depending on \( x \)?
5. How would the area expression change if both dimensions were reduced by half?

**Tip:** Always remember to apply the distributive property carefully when multiplying binomials to avoid errors.

Which expression represents the area, in square units, of the rectangle?

Learn how to calculate the area of a rectangle with binomial expressions for length and width. This example demonstrates how to apply the distributive property to find the correct quadratic expression, suitable for Grades 7-10.

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