To find the area of the rectangle, given its length and width, we use the formula for the area of a rectangle:

$A = \text{Length} \times \text{Width}$

Given:

• Length, $L = 4x + 9$
• Width, $W = 5x - 7$

Step 1: Multiply $L$ and $W$

$A = (4x + 9)(5x - 7)$

Step 2: Use the distributive property (FOIL method) to expand this expression

$A = (4x)(5x) + (4x)(-7) + (9)(5x) + (9)(-7)$

Step 3: Simplify each term

1. $(4x)(5x) = 20x^2$
2. $(4x)(-7) = -28x$
3. $(9)(5x) = 45x$
4. $(9)(-7) = -63$

Step 4: Combine like terms

$A = 20x^2 + (-28x + 45x) - 63$ $A = 20x^2 + 17x - 63$

Final Answer:

The area of the rectangle is:

$\boxed{20x^2 + 17x - 63}$

This corresponds to the option on the orange tile.

Would you like further details, or have any questions?

Here are five related questions:

1. How do we determine which terms to combine in the expansion?
2. What is the significance of using the distributive property in polynomial multiplication?
3. How do we verify the final area expression through substitution?
4. How would the result change if either the length or width expression was different?
5. How do we interpret negative terms in the context of area?

Tip: When expanding binomials, carefully handle each term, especially with signs, to avoid calculation errors.