`9a^(2)-(2b-c)^(2)`=_______

Options

A. `(3a-2b-c)(3a+2b-c)`
B. `(3a+2b-c)(3a-2b+c)`
C. `(-3a+2b+c)(3a+2b-c)`
D. `(3a+2b+c)(3a+2b-c)`

Answer

To factor the expression \( 9a^2 - (2b - c)^2 \), we can use the difference of squares formula, which states that \( x^2 - y^2 = (x - y)(x + y) \).

### Step-by-step Solution:

1. **Identify the squares**:
We can rewrite \( 9a^2 \) as \( (3a)^2 \) and \( (2b - c)^2 \) is already in square form.
\[ 9a^2 - (2b - c)^2 = (3a)^2 - (2b - c)^2 \]

2. **Apply the difference of squares formula**:
Using the formula \( x^2 - y^2 = (x - y)(x + y) \), we set \( x = 3a \) and \( y = (2b - c) \).
\[ = (3a - (2b - c))(3a + (2b - c)) \]

3. **Simplify the expressions**:
Now, simplify both factors:
- For the first factor:
\[ 3a - (2b - c) = 3a - 2b + c \]
- For the second factor:
\[ 3a + (2b - c) = 3a + 2b - c \]

4. **Combine the factors**:
Now we can write the final factored form:
\[ = (3a - 2b + c)(3a + 2b - c) \]

Thus, the factorized form of \( 9a^2 - (2b - c)^2 \) is:
\[ (3a - 2b + c)(3a + 2b - c) \]