The factors of
`(12 x - 8y) (a - 2b) + ( 6b - 2a) ( 3x - 2y)` are :

Options

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

Answer

To factor the expression \((12x - 8y)(a - 2b) + (6b - 2a)(3x - 2y)\), we will follow these steps:

### Step 1: Identify Common Factors
First, we notice that both terms of the expression have a common factor. The first term is \((12x - 8y)(a - 2b)\) and the second term is \((6b - 2a)(3x - 2y)\).

### Step 2: Factor Out Common Terms
We can factor out the common term \((3x - 2y)\) from both parts of the expression.

\[ (12x - 8y)(a - 2b) + (6b - 2a)(3x - 2y) = (3x - 2y) \left( \frac{(12x - 8y)(a - 2b)}{(3x - 2y)} + \frac{(6b - 2a)(3x - 2y)}{(3x - 2y)} \right) \]

### Step 3: Simplify the Expression
Now we simplify the remaining expression.

1. The first term simplifies to \(4(a - 2b)\) because \(12x - 8y = 4(3x - 2y)\).
2. The second term simplifies to \((6b - 2a)\).

So, we have:

\[ (3x - 2y) \left( 4(a - 2b) + (6b - 2a) \right) \]

### Step 4: Combine Like Terms
Now we combine the terms inside the parentheses:

\[ 4a - 8b + 6b - 2a = (4a - 2a) + (-8b + 6b) = 2a - 2b \]

### Step 5: Factor Out the Common Factor
Now we can factor out the common factor of 2:

\[ 2(a - b) \]

### Final Expression
Putting it all together, we have:

\[ (3x - 2y)(2(a - b)) \]

Thus, the factors of the given expression are:

\[ 2(3x - 2y)(a - b) \]