$$ \frac{\left(x^5-3x^3+6x^2+9x+6\right)}{\left(x+2\right)} $$

To divide the polynomial $x^5 - 3x^3 + 6x^2 + 9x + 6$ by $x + 2$, we can use either long division or synthetic division. We'll use synthetic division since the divisor is a binomial of the form $x + a$.

### Step 1: Set up synthetic division

The divisor is $x + 2$, so we use $-2$ in synthetic division. Write down the coefficients of the polynomial $x^5 - 3x^3 + 6x^2 + 9x + 6$:

- Coefficients: $1, 0, -3, 6, 9, 6$

### Step 2: Perform synthetic division

1. **Bring down the first coefficient**:
   - The first coefficient is $1$, so we bring it down as it is.

2. **Multiply by $-2$ and add to the next coefficient**:
   - $1 \times -2 = -2$
   - Add this to the next coefficient $0$: $0 + (-2) = -2$

3. **Repeat the process** for each coefficient:
   - $-2 \times -2 = 4$
   - Add this to the next coefficient $-3$: $-3 + 4 = 1$

   - $1 \times -2 = -2$
   - Add this to the next coefficient $6$: $6 + (-2) = 4$

   - $4 \times -2 = -8$
   - Add this to the next coefficient $9$: $9 + (-8) = 1$

   - $1 \times -2 = -2$
   - Add this to the next coefficient $6$: $6 + (-2) = 4$

### Step 3: Write the quotient and remainder

The coefficients of the quotient are $1, -2, 1, 4, 1$, and the remainder is $4$.

So, the result of the division is:
\[
\frac{x^5 - 3x^3 + 6x^2 + 9x + 6}{x + 2} = x^4 - 2x^3 + x^2 + 4x + 1 + \frac{4}{x+2}
\]

Would you like to dive deeper into any part of this process, or do you have any questions?

Here are five related questions for further exploration:
1. How can you verify the result of a polynomial division?
2. What are the benefits of using synthetic division over long division?
3. How would you divide a polynomial by a binomial not in the form $x + a$?
4. Can you describe how synthetic division works for divisors with a leading coefficient other than 1?
5. What would be the result of dividing $x^5 - 3x^3 + 6x^2 + 9x + 6$ by $x - 1$?

**Tip:** Always double-check your division by multiplying the quotient by the divisor and adding the remainder to ensure you get back the original polynomial.

Learn how to divide the polynomial x^5 - 3x^3 + 6x^2 + 9x + 6 by x + 2 using synthetic division. Detailed step-by-step solution provided. Suitable for advanced high school students studying polynomial division.

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