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This image appears to show a long division of polynomials process. It looks like you're dividing \( 3x^3 - 23x^2 - 17x - 6 \) by \( 3x + 1 \). Let's walk through the polynomial division step by step.

### Step-by-Step Explanation:

1. **First division step:**
   \[
   \frac{3x^3}{3x} = x^2
   \]
   Now, multiply \( x^2 \) by \( 3x + 1 \):
   \[
   x^2(3x + 1) = 3x^3 + x^2
   \]
   Subtract this from the original polynomial:
   \[
   (3x^3 - 23x^2 - 17x - 6) - (3x^3 + x^2) = -24x^2 - 17x - 6
   \]

2. **Second division step:**
   \[
   \frac{-24x^2}{3x} = -8x
   \]
   Multiply \( -8x \) by \( 3x + 1 \):
   \[
   -8x(3x + 1) = -24x^2 - 8x
   \]
   Subtract:
   \[
   (-24x^2 - 17x - 6) - (-24x^2 - 8x) = -9x - 6
   \]

3. **Third division step:**
   \[
   \frac{-9x}{3x} = -3
   \]
   Multiply \( -3 \) by \( 3x + 1 \):
   \[
   -3(3x + 1) = -9x - 3
   \]
   Subtract:
   \[
   (-9x - 6) - (-9x - 3) = -3
   \]

So, the quotient is:
\[
x^2 - 8x - 3
\]
And the remainder is \( -3 \).

### Final Answer:
\[
\frac{3x^3 - 23x^2 - 17x - 6}{3x + 1} = x^2 - 8x - 3 \quad \text{with remainder} \, -3
\]

Would you like a deeper explanation on any specific step?

### Relative Questions:
1. How do you handle division when the remainder is non-zero?
2. What is the difference between synthetic division and long division for polynomials?
3. How can you verify the result of a polynomial division?
4. What happens when the divisor is of a higher degree than the dividend?
5. How would the process change if there were missing terms in the dividend?

### Tip:
When dividing polynomials, always arrange terms in descending order of degree and ensure that all powers of the variable are present, even if their coefficients are zero.

Perform the polynomial long division of (3x^3 - 23x^2 - 17x - 6) by (3x + 1).

Step-by-step guide on how to perform polynomial long division of (3x^3 - 23x^2 - 17x - 6) by (3x + 1). Learn key algebraic concepts and the division algorithm for polynomials, suitable for high school students in Grades 9-12.

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