To divide the polynomial \(3x^4 - x^3 + x - 2\) by \(3x^2 + x + 1\), we can use polynomial long division. Here's a step-by-step solution:

### Step-by-step Polynomial Long Division

1. **Set Up the Division:**
   We are dividing the dividend \(3x^4 - x^3 + x - 2\) by the divisor \(3x^2 + x + 1\).

2. **Divide the Leading Terms:**
   Divide the leading term of the dividend \(3x^4\) by the leading term of the divisor \(3x^2\):

   \[
   \frac{3x^4}{3x^2} = x^2
   \]

   This is the first term of the quotient.

3. **Multiply and Subtract:**
   Multiply \(x^2\) by the divisor \(3x^2 + x + 1\):

   \[
   x^2(3x^2 + x + 1) = 3x^4 + x^3 + x^2
   \]

   Subtract this from the original dividend:

   \[
   (3x^4 - x^3 + x - 2) - (3x^4 + x^3 + x^2) = -2x^3 - x^2 + x - 2
   \]

4. **Repeat the Process:**
   Now divide the new leading term \(-2x^3\) by \(3x^2\):

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

   This is the next term in the quotient.

   Multiply \(-\frac{2}{3}x\) by the divisor:

   \[
   -\frac{2}{3}x(3x^2 + x + 1) = -2x^3 - \frac{2}{3}x^2 - \frac{2}{3}x
   \]

   Subtract this from the current remainder:

   \[
   (-2x^3 - x^2 + x - 2) - (-2x^3 - \frac{2}{3}x^2 - \frac{2}{3}x) = -\frac{1}{3}x^2 + \frac{5}{3}x - 2
   \]

5. **Continue the Process:**
   Divide \(-\frac{1}{3}x^2\) by \(3x^2\):

   \[
   \frac{-\frac{1}{3}x^2}{3x^2} = -\frac{1}{9}
   \]

   This is the next term in the quotient.

   Multiply \(-\frac{1}{9}\) by the divisor:

   \[
   -\frac{1}{9}(3x^2 + x + 1) = -\frac{1}{3}x^2 - \frac{1}{9}x - \frac{1}{9}
   \]

   Subtract this from the current remainder:

   \[
   \left(-\frac{1}{3}x^2 + \frac{5}{3}x - 2\right) - \left(-\frac{1}{3}x^2 - \frac{1}{9}x - \frac{1}{9}\right) = \frac{16}{9}x - \frac{17}{9}
   \]

6. **Find the Final Quotient and Remainder:**
   Now we have completed the division process. The quotient is:

   \[
   x^2 - \frac{2}{3}x - \frac{1}{9}
   \]

   And the remainder is:

   \[
   \frac{16}{9}x - \frac{17}{9}
   \]

### Final Answer:

\[
\frac{3x^4 - x^3 + x - 2}{3x^2 + x + 1} = x^2 - \frac{2}{3}x - \frac{1}{9} + \frac{\frac{16}{9}x - \frac{17}{9}}{3x^2 + x + 1}
\]

Would you like more details or have any questions?

### Related Questions:

1. How do you perform synthetic division with polynomials?
2. How do you simplify the remainder obtained in polynomial long division?
3. Can you explain the Remainder Theorem and how it applies to polynomial division?
4. What are the applications of polynomial division in real-world scenarios?
5. How do you divide polynomials using synthetic division when the divisor is not linear?

**Tip:** When dividing polynomials, always align terms by their degrees for clarity and accuracy.

Understand how to perform polynomial long division for the expression 3x⁴ - x³ + x - 2 divided by 3x² + x + 1 with a comprehensive step-by-step solution. This guide covers key algebraic concepts, including polynomial division and the Remainder Theorem, suitable for high school students in Grades 9-12.

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