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.