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The problem asks to use synthetic division to divide \( x^4 - x^3 + x^2 - 8x + 3 \) by \( x - 2 \).

### Step 1: Set up synthetic division
Since we are dividing by \( x - 2 \), the divisor is 2.

The coefficients of the polynomial \( x^4 - x^3 + x^2 - 8x + 3 \) are:
\[ [1, -1, 1, -8, 3] \]

### Step 2: Perform synthetic division

1. **Write the divisor (2) on the left and the coefficients on the right:**

\[
\begin{array}{r|rrrrr}
2 & 1 & -1 & 1 & -8 & 3 \\
\end{array}
\]

2. **Bring down the first coefficient (1):**

\[
\begin{array}{r|rrrrr}
2 & 1 & -1 & 1 & -8 & 3 \\
  &   & 2 \\
\hline
  & 1 & 
\end{array}
\]

3. **Multiply 2 by the value just written (1) and write it under the next coefficient (-1):**

\[
\begin{array}{r|rrrrr}
2 & 1 & -1 & 1 & -8 & 3 \\
  &   & 2 & \\
\hline
  & 1 & 1 & 
\end{array}
\]

4. **Add the column:**
\[
-1 + 2 = 1
\]

5. **Multiply 2 by 1 and write it under the next coefficient (1):**
\[
\begin{array}{r|rrrrr}
2 & 1 & -1 & 1 & -8 & 3 \\
  &   & 2 & 2 & \\
\hline
  & 1 & 1 & 3 & 
\end{array}
\]

6. **Add the column:**
\[
1 + 2 = 3
\]

7. **Multiply 2 by 3 and write it under the next coefficient (-8):**
\[
\begin{array}{r|rrrrr}
2 & 1 & -1 & 1 & -8 & 3 \\
  &   & 2 & 2 & 6 & \\
\hline
  & 1 & 1 & 3 & -2 &
\end{array}
\]

8. **Add the column:**
\[
-8 + 6 = -2
\]

9. **Multiply 2 by -2 and write it under the next coefficient (3):**
\[
\begin{array}{r|rrrrr}
2 & 1 & -1 & 1 & -8 & 3 \\
  &   & 2 & 2 & 6 & -4 \\
\hline
  & 1 & 1 & 3 & -2 & -1
\end{array}
\]

10. **Add the column:**
\[
3 + (-4) = -1
\]

### Step 3: Interpret the result
The synthetic division gives:
\[
x^3 + x^2 + 3x - 2 + \frac{-1}{x - 2}
\]

This corresponds to **option D**.

### Final Answer:
The correct option is **D**.

Would you like to go over the synthetic division steps in more detail or have any specific questions?

Here are 5 related questions to think about:
1. How does synthetic division differ from long division of polynomials?
2. What happens when dividing a polynomial by a binomial where the leading coefficient isn't 1?
3. How can synthetic division be used to find roots of polynomials?
4. What is the significance of the remainder in polynomial division?
5. How would the process change if the divisor was \(x + 2\) instead of \(x - 2\)?

**Tip**: Always ensure that the polynomial is in standard form and includes any missing terms with zero coefficients.

Use synthetic division to divide x^4 - x^3 + x^2 - 8x + 3 by x - 2.

Learn how to perform synthetic division on the polynomial x^4 - x^3 + x^2 - 8x + 3 divided by x - 2. Step-by-step instructions show how to find the quotient and remainder using synthetic division.

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