https://file.aiquickdraw.com/mathbot/file/1725232688665lq27vgcr.jpg

The expression given in the image is:

\[
\frac{12x^3 - 13x^2 - 3x + 2}{x + 2}
\]

To solve this using long division, let's work through the process step by step.

### Step 1: Set up the division

We are dividing \(12x^3 - 13x^2 - 3x + 2\) by \(x + 2\). In long division, we write it as:

\[
\begin{array}{r|l}
x + 2 & 12x^3 - 13x^2 - 3x + 2
\end{array}
\]

### Step 2: Divide the leading term
Divide the leading term of the dividend by the leading term of the divisor:

\[
\frac{12x^3}{x} = 12x^2
\]

Place \(12x^2\) above the long division bar:

\[
\begin{array}{r|l}
x + 2 & 12x^3 - 13x^2 - 3x + 2 \\
      & 12x^2
\end{array}
\]

### Step 3: Multiply and subtract
Multiply \(12x^2\) by the entire divisor \(x + 2\) and subtract the result from the dividend:

\[
12x^2 \cdot (x + 2) = 12x^3 + 24x^2
\]

Subtract:

\[
(12x^3 - 13x^2 - 3x + 2) - (12x^3 + 24x^2) = -37x^2 - 3x + 2
\]

### Step 4: Repeat the process
Now, divide the new leading term by the leading term of the divisor:

\[
\frac{-37x^2}{x} = -37x
\]

Place \(-37x\) above the long division bar:

\[
\begin{array}{r|l}
x + 2 & 12x^3 - 13x^2 - 3x + 2 \\
      & 12x^2 - 37x
\end{array}
\]

Multiply and subtract again:

\[
-37x \cdot (x + 2) = -37x^2 - 74x
\]

Subtract:

\[
(-37x^2 - 3x + 2) - (-37x^2 - 74x) = 71x + 2
\]

### Step 5: Continue the process
Now, divide the new leading term by the leading term of the divisor:

\[
\frac{71x}{x} = 71
\]

Place \(71\) above the long division bar:

\[
\begin{array}{r|l}
x + 2 & 12x^3 - 13x^2 - 3x + 2 \\
      & 12x^2 - 37x + 71
\end{array}
\]

Multiply and subtract:

\[
71 \cdot (x + 2) = 71x + 142
\]

Subtract:

\[
(71x + 2) - (71x + 142) = -140
\]

### Step 6: Write the final answer
The quotient is \(12x^2 - 37x + 71\) and the remainder is \(-140\). Therefore, the division can be expressed as:

\[
\frac{12x^3 - 13x^2 - 3x + 2}{x + 2} = 12x^2 - 37x + 71 - \frac{140}{x + 2}
\]

Would you like more details on any part of this solution? Here are some related questions that might interest you:

1. What is synthetic division and how does it compare to long division?
2. How can the remainder theorem be used to check the remainder in polynomial division?
3. Can you explain how to handle division when the divisor is a polynomial of higher degree?
4. What are some common mistakes to avoid when performing polynomial long division?
5. How do you interpret the remainder in the context of polynomial division?

**Tip:** Always check your work by multiplying the quotient by the divisor and adding the remainder to ensure it matches the original polynomial.

Learn how to perform long division on the polynomial 12x^3 - 13x^2 - 3x + 2 by the divisor x + 2. This detailed solution covers the process of polynomial division step-by-step, suitable for high school students and anyone interested in advanced algebraic techniques.

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