Which of the following is correct ?

Options

A. `(4x + 5y) (5x - 4y) = 20 x ^(2) + 9xy - 20 y ^(2)`
B. `(4x+ 5y) (5x - 4y) = 20 x ^(2) + 41 xy - 20 y ^(2)`
C. `(4x + 5y) (5x - 4y) = 20 x ^(2) - 9xy + 20 y ^(2)`
D. `(4x + 5y) ( 5x - 4y) = 20 x ^(2) - 41 xy + 20 y ^(2)`

Answer

To solve the problem, we need to expand the expression given in the options and identify which one is correct. The expression we are working with is (4x + 5y)(5x - 4y).

### Step-by-Step Solution:

1. **Write down the expression**:
\[ (4x + 5y)(5x - 4y) \]

2. **Use the distributive property (FOIL method)**:
- First, multiply the first terms:
\[ 4x \cdot 5x = 20x^2 \]
- Next, multiply the outer terms:
\[ 4x \cdot (-4y) = -16xy \]
- Then, multiply the inner terms:
\[ 5y \cdot 5x = 25xy \]
- Finally, multiply the last terms:
\[ 5y \cdot (-4y) = -20y^2 \]

3. **Combine all the results**:
\[ 20x^2 - 16xy + 25xy - 20y^2 \]

4. **Combine like terms**:
- Combine the xy terms:
\[ -16xy + 25xy = 9xy \]
- Therefore, the expression simplifies to:
\[ 20x^2 + 9xy - 20y^2 \]

5. **Identify the correct option**:
- Now we compare our result \(20x^2 + 9xy - 20y^2\) with the options given.
- It matches with option A.

### Final Result:
The correct option is:
\[ 20x^2 + 9xy - 20y^2 \]