Divide using synthetic division.
StartFraction x Superscript 5 Baseline minus 2 x Superscript 4 Baseline minus 4 x cubed minus 3 x squared plus x plus 2 Over x plus 2 EndFraction

To divide \( \frac{x^5 - 2x^4 - 4x^3 - 3x^2 + x + 2}{x + 2} \) using synthetic division, follow these steps:

### Step 1: Set up the synthetic division.
We are dividing by \( x + 2 \), so the root of \( x + 2 = 0 \) is \( -2 \). This will be the number we use in synthetic division.

The coefficients of the dividend \( x^5 - 2x^4 - 4x^3 - 3x^2 + x + 2 \) are:
\[ 1, -2, -4, -3, 1, 2 \]

So, we'll use these coefficients with \( -2 \) as the divisor.

### Step 2: Perform synthetic division.

Set up the synthetic division as follows:

\[
\begin{array}{r|rrrrrr}
-2 & 1 & -2 & -4 & -3 & 1 & 2 \\
   &   & -2 & 8  & -8 & 22 & -45 \\
\hline
   & 1 & -4 & 4  & -11 & 23 & -43 \\
\end{array}
\]

### Step 3: Interpret the result.

The bottom row \( 1, -4, 4, -11, 23 \) represents the coefficients of the quotient, while the remainder is \( -43 \). Therefore, the quotient is:

\[
x^4 - 4x^3 + 4x^2 - 11x + 23
\]

And the remainder is \( -43 \).

### Final Answer:
\[
\frac{x^5 - 2x^4 - 4x^3 - 3x^2 + x + 2}{x + 2} = x^4 - 4x^3 + 4x^2 - 11x + 23 - \frac{43}{x + 2}
\]

Would you like further clarification or more details on synthetic division?

Here are 5 related questions:
1. How can synthetic division be used to check if a polynomial is divisible by a binomial?
2. What is the relationship between synthetic division and the remainder theorem?
3. How do you interpret the remainder in synthetic division?
4. What is the degree of the quotient when dividing a degree 5 polynomial by a degree 1 binomial?
5. Can synthetic division be applied when dividing by polynomials of higher degrees?

**Tip:** Always check if the divisor is of the form \( x - c \) before starting synthetic division.

Divide using synthetic division. \( \frac{x^5 - 2x^4 - 4x^3 - 3x^2 + x + 2}{x + 2} \)

Learn how to divide the polynomial x^5 - 2x^4 - 4x^3 - 3x^2 + x + 2 by x + 2 using synthetic division. This problem explores key algebraic concepts like polynomial division, the remainder theorem, and the factor theorem. Detailed steps provided, suitable for students in Grades 10-12.

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