Factorise `(2a + 3b)^(2) - (3a - 2b)^(2)` .

Options

A. `(5a + b) (5a - b)`
B. `(a + 5b) (a - 5b)`
C. `(5a + b) (5b - a)`
D. `(5a + b) (5b + a)`

Explanation

To factorize the expression \((2a + 3b)^{2} - (3a - 2b)^{2}\), we can use the difference of squares formula, which states that \(A^{2} - B^{2} = (A + B)(A - B)\).

### Step-by-Step Solution:

1. **Identify A and B**:
- Let \(A = 2a + 3b\)
- Let \(B = 3a - 2b\)

2. **Apply the Difference of Squares Formula**:
- According to the formula, we have:
\[ (2a + 3b)^{2} - (3a - 2b)^{2} = (A + B)(A - B) \]

3. **Calculate A + B**:
- \(A + B = (2a + 3b) + (3a - 2b)\)
- Combine like terms:
\[ A + B = 2a + 3a + 3b - 2b = 5a + b \]

4. **Calculate A - B**:
- \(A - B = (2a + 3b) - (3a - 2b)\)
- Distributing the negative sign:
\[ A - B = 2a + 3b - 3a + 2b = 2a - 3a + 3b + 2b = -a + 5b = 5b - a \]

5. **Combine the Results**:
- Now substituting back into the formula:
\[ (2a + 3b)^{2} - (3a - 2b)^{2} = (5a + b)(5b - a) \]

### Final Answer:
The factorized form of \((2a + 3b)^{2} - (3a - 2b)^{2}\) is:
\[ (5a + b)(5b - a) \]