Use synthetic division to find the result when 4, x, to the power 4 , plus, 2, x, cubed, minus, 2, x, squared, plus, 23, x, minus, 24x 
4
 +2x 
3
 −2x 
2
 +23x−2 is divided by x, plus, 2x+2. If there is a remainder, express the result in the form q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fractionq(x)+ 
b(x)
r(x)
​	
 .

It looks like you're asking to use **synthetic division** to divide the polynomial \(4x^4 + 2x^3 - 2x^2 + 23x - 24\) by \(x + 2\). Let's go through the process step by step:

### Step 1: Write the coefficients of the polynomial.
The coefficients of \(4x^4 + 2x^3 - 2x^2 + 23x - 24\) are:
\[
4, 2, -2, 23, -24
\]

### Step 2: Set up synthetic division.
Since we're dividing by \(x + 2\), we use the **opposite sign** of the constant term, which is \( -2 \), in the synthetic division process.

So, the setup will look like this:

\[
\begin{array}{r|rrrrr}
-2 & 4 & 2 & -2 & 23 & -24 \\
   &   &   &   &    &    \\
\end{array}
\]

### Step 3: Perform synthetic division.
1. Bring down the first coefficient (4) to the bottom row.

\[
\begin{array}{r|rrrrr}
-2 & 4 & 2 & -2 & 23 & -24 \\
   &   &   &   &    &    \\
   & 4 \\
\end{array}
\]

2. Multiply the number brought down by \(-2\) (which gives \(-8\)) and add it to the next coefficient (2). This gives:

\[
2 + (-8) = -6
\]

3. Repeat the process:

\[
\begin{array}{r|rrrrr}
-2 & 4 & 2 & -2 & 23 & -24 \\
   &   & -8 & 12 & -34 & 68 \\
   & 4 & -6 & 10 & 43 & 44 \\
\end{array}
\]

### Step 4: Interpret the result.
The numbers at the bottom represent the coefficients of the quotient polynomial and the remainder.

The quotient is:
\[
4x^3 - 6x^2 + 10x + 43
\]

The remainder is \(44\).

### Final Answer:
So, the result of dividing \(4x^4 + 2x^3 - 2x^2 + 23x - 24\) by \(x + 2\) is:
\[
4x^3 - 6x^2 + 10x + 43 + \frac{44}{x + 2}
\]

Do you have any questions, or would you like a deeper explanation?

---

### 5 Related Questions:
1. What is the long division method for polynomials, and how does it differ from synthetic division?
2. Can synthetic division be used for divisors that are not linear (e.g., \(x^2 + x + 1\))?
3. How do you handle synthetic division when there are missing terms in the polynomial?
4. What are the applications of synthetic division in real-world problems?
5. How can you verify the result of synthetic division using polynomial multiplication?

### Tip:
When using synthetic division, always check if the divisor is of the form \(x - c\), as synthetic division is designed specifically for such divisors.

Use synthetic division to find the result when 4x^4 + 2x^3 - 2x^2 + 23x - 24 is divided by x + 2. If there is a remainder, express the result in the form q(x) + r(x)/b(x).

Step-by-step guide to using synthetic division to divide the polynomial 4x^4 + 2x^3 - 2x^2 + 23x - 24 by x + 2. Learn how to find the quotient and remainder, with detailed explanations for high school students in Grades 9-12.

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