Divide :
`12x^(2) + 7xy - 12y^(2)` by 3x + 4y

Answer

To divide the polynomial \( 12x^2 + 7xy - 12y^2 \) by \( 3x + 4y \), we will use polynomial long division. Here’s a step-by-step solution:

### Step 1: Set up the division
We write the dividend \( 12x^2 + 7xy - 12y^2 \) under the long division symbol and the divisor \( 3x + 4y \) outside.

### Step 2: Divide the leading terms
We need to determine what to multiply \( 3x \) by to get \( 12x^2 \).
\[ \frac{12x^2}{3x} = 4x \]
So, we will multiply \( 3x + 4y \) by \( 4x \).

### Step 3: Multiply and subtract
Now we multiply \( 4x \) by \( 3x + 4y \):
\[ 4x \cdot (3x + 4y) = 12x^2 + 16xy \]
Now, we subtract this from the original polynomial:
\[ (12x^2 + 7xy - 12y^2) - (12x^2 + 16xy) = (7xy - 16xy) - 12y^2 = -9xy - 12y^2 \]

### Step 4: Repeat the process
Now, we need to divide the leading term of the new polynomial \( -9xy \) by \( 3x \):
\[ \frac{-9xy}{3x} = -3y \]
So, we will multiply \( 3x + 4y \) by \( -3y \).

### Step 5: Multiply and subtract again
Now we multiply \( -3y \) by \( 3x + 4y \):
\[ -3y \cdot (3x + 4y) = -9xy - 12y^2 \]
Now, we subtract this from the polynomial we obtained in the last step:
\[ (-9xy - 12y^2) - (-9xy - 12y^2) = 0 \]

### Conclusion
Since the remainder is \( 0 \), the division is complete. The result of the division is:
\[ 4x - 3y \]

### Final Answer
Thus, the answer to the division of \( 12x^2 + 7xy - 12y^2 \) by \( 3x + 4y \) is:
\[ \boxed{4x - 3y} \]