Which of the following correct ?

Options

A. `(4x +3y) ^(2) = 16 x ^(2)-12 xy + 9 y ^(2)`
B. `(4x + 3y) ^(2) = 16 x ^(2) - 24 xy + 9y ^(2)`
C. `(4x - 3y) ^(2) =16 x ^(2) + 12 xy - 9y ^(2)`
D. `(4x - 3y) ^(2) = 16 x ^(2) - 24 xy + 9y ^(2)`

Answer

To determine which of the given options is correct, we will expand the expressions using the formulas for the square of a binomial.

### Step-by-Step Solution:

1. **Identify the binomial expressions**:
- The expressions we need to evaluate are:
- (1) \( (4x + 3y)^2 \)
- (2) \( (4x - 3y)^2 \)

2. **Use the formula for the square of a binomial**:
- The formula for \( (a + b)^2 \) is \( a^2 + 2ab + b^2 \).
- The formula for \( (a - b)^2 \) is \( a^2 - 2ab + b^2 \).

3. **Expand \( (4x + 3y)^2 \)**:
- Here, \( a = 4x \) and \( b = 3y \).
- Applying the formula:
\[ (4x + 3y)^2 = (4x)^2 + 2(4x)(3y) + (3y)^2 \]
- Calculate each term:
- \( (4x)^2 = 16x^2 \)
- \( 2(4x)(3y) = 24xy \)
- \( (3y)^2 = 9y^2 \)
- Combine the terms:
\[ (4x + 3y)^2 = 16x^2 + 24xy + 9y^2 \]

4. **Expand \( (4x - 3y)^2 \)**:
- Here, \( a = 4x \) and \( b = 3y \).
- Applying the formula:
\[ (4x - 3y)^2 = (4x)^2 - 2(4x)(3y) + (3y)^2 \]
- Calculate each term:
- \( (4x)^2 = 16x^2 \)
- \( -2(4x)(3y) = -24xy \)
- \( (3y)^2 = 9y^2 \)
- Combine the terms:
\[ (4x - 3y)^2 = 16x^2 - 24xy + 9y^2 \]

5. **Compare with the options**:
- Now we compare our results with the given options:
- **Option 1**: \( (4x + 3y)^2 = 16x^2 - 12xy + 9y^2 \) (Incorrect)
- **Option 2**: \( (4x + 3y)^2 = 16x^2 - 24xy + 9y^2 \) (Incorrect)
- **Option 3**: \( (4x - 3y)^2 = 16x^2 + 12xy - 9y^2 \) (Incorrect)
- **Option 4**: \( (4x - 3y)^2 = 16x^2 - 24xy + 9y^2 \) (Correct)

### Conclusion:
The correct option is **Option 4**: \( (4x - 3y)^2 = 16x^2 - 24xy + 9y^2 \).