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Let's solve the synthetic division step by step using the image you've uploaded.

We are tasked with dividing the polynomial:

\[
2x^4 + 4x^3 - 6x^2 + 3x + 8
\]

by \( x + 3 \). This will be done using synthetic division.

### Step 1: Set up the synthetic division

We use the root of \( x + 3 \), which is \( -3 \), as the divisor in synthetic division.

The coefficients of the polynomial are:

\[
2, 4, -6, 3, 8
\]

Now, we will carry out the synthetic division, starting with \( -3 \) as the divisor.

### Step 2: Perform synthetic division

Set up the synthetic division table, placing \( -3 \) on the outside and the coefficients \( 2, 4, -6, 3, 8 \) inside:

\[
\begin{array}{r|rrrrr}
-3 & 2 & 4 & -6 & 3 & 8 \\
   &   &   &   &   &   \\
\hline
   &   &   &   &   &   \\
\end{array}
\]

1. Bring down the first coefficient, \( 2 \).
   
2. Multiply \( 2 \) by \( -3 \), which gives \( -6 \), and write it under the next coefficient \( 4 \).
   
3. Add \( 4 + (-6) = -2 \), and place the result under the line.

4. Multiply \( -2 \) by \( -3 \), giving \( 6 \), and write it under the next coefficient \( -6 \).

5. Add \( -6 + 6 = 0 \).

6. Multiply \( 0 \) by \( -3 \), resulting in \( 0 \), and write it under the next coefficient \( 3 \).

7. Add \( 3 + 0 = 3 \).

8. Multiply \( 3 \) by \( -3 \), giving \( -9 \), and write it under the next coefficient \( 8 \).

9. Add \( 8 + (-9) = -1 \).

Now, the synthetic division table looks like this:

\[
\begin{array}{r|rrrrr}
-3 & 2 & 4 & -6 & 3 & 8 \\
   &   & -6 & 6 & 0 & -9 \\
\hline
   & 2 & -2 & 0 & 3 & -1 \\
\end{array}
\]

### Step 3: Interpret the result

The bottom row \( 2, -2, 0, 3 \) represents the quotient, and the last number, \( -1 \), is the remainder.

Thus, the quotient is:

\[
2x^3 - 2x^2 + 3
\]

And the remainder is:

\[
-1
\]

### Step 4: Write the final answer

We can express the result of the division as:

\[
\frac{2x^4 + 4x^3 - 6x^2 + 3x + 8}{x + 3} = 2x^3 - 2x^2 + 3 + \frac{-1}{x + 3}
\]

This completes the synthetic division process.

Would you like further explanation on any part of this process? Also, here are 5 related questions to expand on this topic:

1. How would the process differ if dividing by \(x - 3\) instead of \(x + 3\)?
2. What is the long division method, and how does it compare to synthetic division?
3. Can synthetic division be used for non-linear divisors like \(x^2 + 1\)?
4. How is the remainder theorem related to synthetic division?
5. How does synthetic division simplify when the leading coefficient is 1?

**Tip**: Always verify the quotient and remainder by multiplying the divisor with the quotient and adding the remainder to check if it results in the original polynomial.

Use synthetic division to find the quotient and remainder when 2x^4 + 4x^3 - 6x^2 + 3x + 8 is divided by x + 3.

Learn how to perform synthetic division to divide the polynomial 2x^4 + 4x^3 - 6x^2 + 3x + 8 by x + 3. This problem covers key algebraic concepts such as polynomial division and the remainder theorem. Step-by-step solution provided for students in Grades 9-12.

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