To simplify the expression \(\frac{x^5 + 5x^2 + x + 2}{x + 1}\), we can perform polynomial long division. Let's go step by step.

1. **Divide the leading term** of \(x^5\) by the leading term of \(x + 1\), which is \(x\):
   \[
   \frac{x^5}{x} = x^4
   \]

2. **Multiply** \(x^4\) by the entire divisor \(x + 1\):
   \[
   x^4(x + 1) = x^5 + x^4
   \]

3. **Subtract** this from the original polynomial:
   \[
   (x^5 + 5x^2 + x + 2) - (x^5 + x^4) = -x^4 + 5x^2 + x + 2
   \]

4. **Divide the leading term** \(-x^4\) by \(x\):
   \[
   \frac{-x^4}{x} = -x^3
   \]

5. **Multiply** \(-x^3\) by \(x + 1\):
   \[
   -x^3(x + 1) = -x^4 - x^3
   \]

6. **Subtract** this from the current polynomial:
   \[
   (-x^4 + 5x^2 + x + 2) - (-x^4 - x^3) = x^3 + 5x^2 + x + 2
   \]

7. **Divide the leading term** \(x^3\) by \(x\):
   \[
   \frac{x^3}{x} = x^2
   \]

8. **Multiply** \(x^2\) by \(x + 1\):
   \[
   x^2(x + 1) = x^3 + x^2
   \]

9. **Subtract** this from the current polynomial:
   \[
   (x^3 + 5x^2 + x + 2) - (x^3 + x^2) = 4x^2 + x + 2
   \]

10. **Divide the leading term** \(4x^2\) by \(x\):
    \[
    \frac{4x^2}{x} = 4x
    \]

11. **Multiply** \(4x\) by \(x + 1\):
    \[
    4x(x + 1) = 4x^2 + 4x
    \]

12. **Subtract** this from the current polynomial:
    \[
    (4x^2 + x + 2) - (4x^2 + 4x) = -3x + 2
    \]

13. **Divide the leading term** \(-3x\) by \(x\):
    \[
    \frac{-3x}{x} = -3
    \]

14. **Multiply** \(-3\) by \(x + 1\):
    \[
    -3(x + 1) = -3x - 3
    \]

15. **Subtract** this from the current polynomial:
    \[
    (-3x + 2) - (-3x - 3) = 5
    \]

Now, the quotient is:
\[
x^4 - x^3 + x^2 + 4x - 3
\]
And the remainder is \(5\).

Thus, the expression simplifies to:
\[
\frac{x^5 + 5x^2 + x + 2}{x + 1} = x^4 - x^3 + x^2 + 4x - 3 + \frac{5}{x + 1}
\]

Would you like more details or have any questions about the process?

Here are five related questions:
1. How do you determine the quotient in polynomial division?
2. What are the steps to check if your division is correct?
3. Can synthetic division be used for this problem?
4. What are the conditions for a remainder in polynomial division to be zero?
5. How do you factor polynomials to simplify division?

**Tip:** Always check for common factors between the numerator and denominator before dividing, as factoring can simplify the process.

This page provides a detailed step-by-step solution for dividing the polynomial (x^5 + 5x^2 + x + 2) by (x + 1) using polynomial long division. The process includes identifying the quotient and remainder, and is suitable for students in Grades 9-12 studying algebra and polynomial operations.

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