What number should be added to `2x^3 - 3x^2 - 8x + 3` so that the resulting polynomial leaves the remainder 10 when divided by 2x + 1 ?

Answer

To solve the problem, we need to determine what number should be added to the polynomial \(2x^3 - 3x^2 - 8x + 3\) so that when it is divided by \(2x + 1\), the remainder is 10.

### Step-by-Step Solution:

1. **Identify the Polynomial and the Divisor**:
The polynomial is \(P(x) = 2x^3 - 3x^2 - 8x + 3\) and the divisor is \(D(x) = 2x + 1\).

2. **Use the Remainder Theorem**:
According to the Remainder Theorem, the remainder of the division of a polynomial \(P(x)\) by a linear divisor \(D(x) = ax + b\) can be found by evaluating \(P\left(-\frac{b}{a}\right)\). Here, \(a = 2\) and \(b = 1\), so we need to evaluate \(P\left(-\frac{1}{2}\right)\).

3. **Calculate \(P\left(-\frac{1}{2}\right)\)**:
\[ P\left(-\frac{1}{2}\right) = 2\left(-\frac{1}{2}\right)^3 - 3\left(-\frac{1}{2}\right)^2 - 8\left(-\frac{1}{2}\right) + 3 \]
\[ = 2\left(-\frac{1}{8}\right) - 3\left(\frac{1}{4}\right) + 4 + 3 \]
\[ = -\frac{1}{4} - \frac{3}{4} + 4 + 3 \]
\[ = -1 + 7 = 6 \]

4. **Determine the Required Addition**:
We found that the remainder when \(P(x)\) is divided by \(2x + 1\) is 6. We want the remainder to be 10. Therefore, we need to find the number \(k\) such that:
\[ 6 + k = 10 \]
\[ k = 10 - 6 = 4 \]

5. **Conclusion**:
The number that should be added to the polynomial \(2x^3 - 3x^2 - 8x + 3\) is \(4\).

### Final Answer:
The number that should be added is \(4\).