Which of the following polynomials, when divided by `3x+4`, equals `2x^(2)+5x-3` with remainder 3 ?

Options

A. `6x^(3)+23x^(2)-11x-12`
B. `6x^(3)+23x^(2)-11x-9`
C. `6x^(3)+23x^(2)-11x-15`
D. `6x^(3)+23x^(2)+11x-9`

Explanation

To find the polynomial that, when divided by \(3x + 4\), equals \(2x^2 + 5x - 3\) with a remainder of 3, we can use the formula for polynomial division:

\[ \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \]

### Step-by-Step Solution:

1. **Identify the components**:
- Divisor: \(3x + 4\)
- Quotient: \(2x^2 + 5x - 3\)
- Remainder: \(3\)

2. **Write the formula for the dividend**:
\[ \text{Dividend} = (3x + 4)(2x^2 + 5x - 3) + 3 \]

3. **Multiply the divisor by the quotient**:
- First, distribute \(3x\) across \(2x^2 + 5x - 3\):
\[ 3x \cdot 2x^2 = 6x^3 \]
\[ 3x \cdot 5x = 15x^2 \]
\[ 3x \cdot (-3) = -9x \]
- Next, distribute \(4\) across \(2x^2 + 5x - 3\):
\[ 4 \cdot 2x^2 = 8x^2 \]
\[ 4 \cdot 5x = 20x \]
\[ 4 \cdot (-3) = -12 \]

4. **Combine the results**:
- Combine the terms from the multiplication:
\[ 6x^3 + (15x^2 + 8x^2) + (-9x + 20x) + (-12) \]
- This simplifies to:
\[ 6x^3 + 23x^2 + 11x - 12 \]

5. **Add the remainder**:
- Now add the remainder \(3\) to the polynomial:
\[ 6x^3 + 23x^2 + 11x - 12 + 3 = 6x^3 + 23x^2 + 11x - 9 \]

6. **Final result**:
- The polynomial that satisfies the condition is:
\[ \text{Dividend} = 6x^3 + 23x^2 + 11x - 9 \]

### Final Answer:
The polynomial is \(6x^3 + 23x^2 + 11x - 9\).

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