You want to divide the polynomial \(-x^3 + 5x^2 - x + 1\) by \(x + 2\). To do this, we will use polynomial long division.

### Step-by-step division:

1. **Divide the leading term:**  
   \[
   \frac{-x^3}{x} = -x^2
   \]

2. **Multiply the divisor \((x + 2)\) by \(-x^2\):**  
   \[
   -x^2(x + 2) = -x^3 - 2x^2
   \]

3. **Subtract the result from the original polynomial:**  
   \[
   (-x^3 + 5x^2 - x + 1) - (-x^3 - 2x^2) = 7x^2 - x + 1
   \]

4. **Repeat the process with the new leading term:**  
   \[
   \frac{7x^2}{x} = 7x
   \]

5. **Multiply the divisor \((x + 2)\) by \(7x\):**  
   \[
   7x(x + 2) = 7x^2 + 14x
   \]

6. **Subtract the result from the current polynomial:**  
   \[
   (7x^2 - x + 1) - (7x^2 + 14x) = -15x + 1
   \]

7. **Divide the new leading term:**  
   \[
   \frac{-15x}{x} = -15
   \]

8. **Multiply the divisor \((x + 2)\) by \(-15\):**  
   \[
   -15(x + 2) = -15x - 30
   \]

9. **Subtract the result from the current polynomial:**  
   \[
   (-15x + 1) - (-15x - 30) = 31
   \]

So, the quotient is:
\[
-q(x) = -x^2 + 7x - 15
\]
with a remainder of 31. 

Thus, the result of the division is:
\[
-x^3 + 5x^2 - x + 1 \div (x + 2) = -x^2 + 7x - 15 + \frac{31}{x+2}
\]

Would you like further clarification or more details?

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**Related Questions:**
1. How do you perform synthetic division?
2. Can polynomial division be done using factoring?
3. What are the uses of polynomial division in calculus?
4. How do you find the remainder in polynomial division without dividing?
5. What happens when you divide polynomials with higher degrees?

**Tip:** Always check your division by multiplying the quotient and divisor and adding the remainder to verify the original polynomial!

Explore how to divide the polynomial -x³ + 5x² - x + 1 by x + 2 using long division. This detailed guide walks through each step, explaining key algebraic concepts like polynomial division and the remainder theorem. Perfect for high school students in Grades 10-12.

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