To perform the division of \( x^4 + 5x^3 + 6x^2 - x - 2 \) by \( x + 2 \), we'll use polynomial long division.

1. **Divide the first term**: Divide the leading term of the dividend \( x^4 \) by the leading term of the divisor \( x \), which gives \( x^3 \).

2. **Multiply**: Multiply \( x^3 \) by the entire divisor \( x + 2 \), resulting in \( x^4 + 2x^3 \).

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

4. **Repeat**: Repeat the process with the new polynomial \( 3x^3 + 6x^2 - x - 2 \):
   - Divide \( 3x^3 \) by \( x \) to get \( 3x^2 \).
   - Multiply \( 3x^2 \) by \( x + 2 \) to get \( 3x^3 + 6x^2 \).
   - Subtract \( (3x^3 + 6x^2) \) from \( 3x^3 + 6x^2 - x - 2 \), which leaves \( -x - 2 \).

5. **Continue**: Continue the process with \( -x - 2 \):
   - Divide \( -x \) by \( x \) to get \( -1 \).
   - Multiply \( -1 \) by \( x + 2 \) to get \( -x - 2 \).
   - Subtract \( -x - 2 \) from \( -x - 2 \), which leaves 0.

The remainder is 0, meaning the division is exact. 

### Final Answer:
\[
\frac{x^4 + 5x^3 + 6x^2 - x - 2}{x + 2} = x^3 + 3x^2 + 0x - 1
\]

Do you want more details or have any questions?

Here are 5 related questions:
1. How do you perform synthetic division with polynomials?
2. What happens when the remainder is not zero in polynomial division?
3. Can you explain the Remainder Theorem?
4. How can you factor polynomials using division?
5. How does polynomial long division relate to the Division Algorithm?

**Tip:** Always align terms of similar degrees vertically when performing polynomial long division to avoid mistakes.

Learn how to perform polynomial long division for x^4 + 5x^3 + 6x^2 - x - 2 divided by x + 2 with a detailed step-by-step solution. This problem covers polynomial division concepts and the Division Algorithm theorem. Suitable for advanced high school students.

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