On dividing `6x^(3)+8x^(2)-3x+8` by a polynomial g(x), the quotient and remainder were `3x+4` and `6x+20`, respectively. Find g(x)

Options

A. `2x-3`
B. `2x^(2)+4`
C. `3x^(2)-4`
D. `2x^(2)-3`

Answer

To find the polynomial \( g(x) \) given the polynomial \( f(x) = 6x^3 + 8x^2 - 3x + 8 \), the quotient \( q(x) = 3x + 4 \), and the remainder \( r(x) = 6x + 20 \), we can use the polynomial division relationship:

\[ f(x) = g(x) \cdot q(x) + r(x) \]

### Step-by-Step Solution:

1. **Write down the relationship**:
\[ f(x) = g(x) \cdot q(x) + r(x) \]

2. **Substitute the known values**:
\[ 6x^3 + 8x^2 - 3x + 8 = g(x) \cdot (3x + 4) + (6x + 20) \]

3. **Rearrange the equation**:
\[ g(x) \cdot (3x + 4) = (6x^3 + 8x^2 - 3x + 8) - (6x + 20) \]

4. **Simplify the right-hand side**:
\[ g(x) \cdot (3x + 4) = 6x^3 + 8x^2 - 3x + 8 - 6x - 20 \]
\[ g(x) \cdot (3x + 4) = 6x^3 + 8x^2 - 9x - 12 \]

5. **Divide both sides by \( (3x + 4) \)** to isolate \( g(x) \):
\[ g(x) = \frac{6x^3 + 8x^2 - 9x - 12}{3x + 4} \]

6. **Perform polynomial long division**:
- Divide the leading term \( 6x^3 \) by \( 3x \) to get \( 2x^2 \).
- Multiply \( 2x^2 \) by \( (3x + 4) \) to get \( 6x^3 + 8x^2 \).
- Subtract this from \( 6x^3 + 8x^2 - 9x - 12 \) to get:
\[ (6x^3 + 8x^2 - 9x - 12) - (6x^3 + 8x^2) = -9x - 12 \]
- Now, divide \( -9x \) by \( 3x \) to get \( -3 \).
- Multiply \( -3 \) by \( (3x + 4) \) to get \( -9x - 12 \).
- Subtract this from \( -9x - 12 \) to get \( 0 \).

7. **Final result**:
\[ g(x) = 2x^2 - 3 \]

### Conclusion:
The polynomial \( g(x) \) is:
\[ g(x) = 2x^2 - 3 \]