The quotient obtained when `3x^4 -x^3 +2x^2 -2x -4 ` is divided by ` x^2 -4` is

Options

A. `3X^2 -7X +16`
B. ` 3x^2 +x +14`
C. `3x^2 -x-14`
D. `3x^2 -x+14`

Answer

To find the quotient obtained when dividing the polynomial \(3x^4 - x^3 + 2x^2 - 2x - 4\) by \(x^2 - 4\), we will use polynomial long division.

### Step-by-Step Solution:

1. **Set up the division**: We write the dividend \(3x^4 - x^3 + 2x^2 - 2x - 4\) and the divisor \(x^2 - 4\).

2. **Divide the leading terms**: Divide the leading term of the dividend \(3x^4\) by the leading term of the divisor \(x^2\):
\[ \frac{3x^4}{x^2} = 3x^2 \]
This is the first term of the quotient.

3. **Multiply and subtract**: Multiply \(3x^2\) by the entire divisor \(x^2 - 4\):
\[ 3x^2(x^2 - 4) = 3x^4 - 12x^2 \]
Now subtract this from the original polynomial:
\[ (3x^4 - x^3 + 2x^2 - 2x - 4) - (3x^4 - 12x^2) = -x^3 + 2x^2 + 12x^2 - 2x - 4 = -x^3 + 14x^2 - 2x - 4 \]

4. **Repeat the process**: Now, divide the leading term \(-x^3\) by \(x^2\):
\[ \frac{-x^3}{x^2} = -x \]
This is the next term of the quotient.

5. **Multiply and subtract again**: Multiply \(-x\) by \(x^2 - 4\):
\[ -x(x^2 - 4) = -x^3 + 4x \]
Subtract this from the current polynomial:
\[ (-x^3 + 14x^2 - 2x - 4) - (-x^3 + 4x) = 14x^2 - 2x - 4 - 4x = 14x^2 - 6x - 4 \]

6. **Continue the process**: Now divide \(14x^2\) by \(x^2\):
\[ \frac{14x^2}{x^2} = 14 \]
This is the last term of the quotient.

7. **Final multiplication and subtraction**: Multiply \(14\) by \(x^2 - 4\):
\[ 14(x^2 - 4) = 14x^2 - 56 \]
Subtract this from the current polynomial:
\[ (14x^2 - 6x - 4) - (14x^2 - 56) = -6x + 52 \]

8. **Conclusion**: The quotient is:
\[ 3x^2 - x + 14 \]
and the remainder is:
\[ -6x + 52 \]

### Final Answer:
The quotient obtained when \(3x^4 - x^3 + 2x^2 - 2x - 4\) is divided by \(x^2 - 4\) is:
\[ \boxed{3x^2 - x + 14} \]