Use synthetic division to find the quotient and remainder when 

x Superscript 5 Baseline minus 2 x cubed plus 2 xx5−2x3+2x 

is divided by 

x plus 1x+1.

To use synthetic division to find the quotient and remainder when dividing \(x^5 - 2x^3 + 2x\) by \(x + 1\), follow these steps:

1. **Set up the synthetic division.**
   - The divisor \(x + 1\) gives a root of \(-1\).
   - Write the coefficients of the dividend \(x^5 - 2x^3 + 2x\), making sure to include coefficients for all powers of \(x\), including zero coefficients.
     - The polynomial is \(x^5 - 2x^3 + 2x\), so the coefficients are:
       \[ 1, 0, -2, 0, 2, 0 \]

2. **Perform the synthetic division:**
   \[
   \begin{array}{r|rrrrrr}
   -1 & 1 & 0 & -2 & 0 & 2 & 0 \\
      &   & -1 & 1 & -3 & 3 & -5 \\
   \hline
      & 1 & -1 & -1 & -3 & 5 & -5 \\
   \end{array}
   \]

Here's a step-by-step explanation of each row in the table:

- **First row:** The root from the divisor and the coefficients from the dividend polynomial.
- **Second row:** Multiply the last entry of the result row by the root and add to the next coefficient.

### Steps:
1. Bring down the first coefficient, \(1\).
2. Multiply \(1\) by \(-1\) (the root from the divisor), giving \(-1\).
3. Add \(-1\) to the next coefficient \(0\), giving \(-1\).
4. Multiply \(-1\) by \(-1\), giving \(1\).
5. Add \(1\) to the next coefficient \(-2\), giving \(-1\).
6. Multiply \(-1\) by \(-1\), giving \(1\).
7. Add \(1\) to the next coefficient \(0\), giving \(1\).
8. Multiply \(1\) by \(-1\), giving \(-1\).
9. Add \(-1\) to the next coefficient \(2\), giving \(1\).
10. Multiply \(1\) by \(-1\), giving \(-1\).
11. Add \(-1\) to the next coefficient \(0\), giving \(-1\).

### Result:
The bottom row (excluding the last value) gives the coefficients of the quotient polynomial, and the last value is the remainder.

Thus, the quotient is:
\[x^4 - x^3 - x^2 - x + 1\]

And the remainder is:
\[-1\]

### Final Answer:
The quotient is \(x^4 - x^3 - x^2 - x + 1\) and the remainder is \(-1\).

Would you like further details or have any questions?

Here are 5 more questions you might want to ask:
1. Can you explain the steps of synthetic division in more detail?
2. How do you determine the roots of a polynomial equation?
3. What is the difference between synthetic division and long division for polynomials?
4. How can synthetic division be used to find polynomial factors?
5. Can you show an example of synthetic division with a different polynomial?

**Tip:** When performing synthetic division, always double-check your multiplication and addition at each step to avoid errors.

Learn how to use synthetic division to find the quotient and remainder when dividing x^5 - 2x^3 + 2x by x + 1. This problem involves advanced polynomial division techniques and is suitable for advanced high school students or college-level mathematics.

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