Factorize `4 (a-b)^(2) - 9 ( a+b)^(2)`.

Options

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

Answer

To factorize the expression \( 4(a-b)^2 - 9(a+b)^2 \), we can follow these steps:

### Step 1: Recognize the structure
The expression \( 4(a-b)^2 - 9(a+b)^2 \) resembles the difference of squares, which can be expressed as \( A^2 - B^2 = (A - B)(A + B) \). Here, we can set:
- \( A = 2(a-b) \)
- \( B = 3(a+b) \)

### Step 2: Apply the difference of squares formula
Now, we can rewrite the expression using the difference of squares:
\[ 4(a-b)^2 - 9(a+b)^2 = (2(a-b))^2 - (3(a+b))^2 = (2(a-b) - 3(a+b))(2(a-b) + 3(a+b)) \]

### Step 3: Simplify each factor
Now we simplify each factor:
1. For \( 2(a-b) - 3(a+b) \):
\[ 2a - 2b - 3a - 3b = -a - 5b = -(a + 5b) \]

2. For \( 2(a-b) + 3(a+b) \):
\[ 2a - 2b + 3a + 3b = 5a + b \]

### Step 4: Combine the factors
Now we can combine the simplified factors:
\[ 4(a-b)^2 - 9(a+b)^2 = (-(a + 5b))(5a + b) \]
This can also be written as:
\[ -(a + 5b)(5a + b) \]

### Final Answer
Thus, the factorized form of the expression \( 4(a-b)^2 - 9(a+b)^2 \) is:
\[ -(a + 5b)(5a + b) \]