To simplify the expression:

\[
\frac{3x^2 - 11x - 4}{3x + 1}
\]

we'll perform polynomial division.

### Step-by-step Solution:

1. **Set Up the Division:**

   Divide the polynomial in the numerator \(3x^2 - 11x - 4\) by the polynomial in the denominator \(3x + 1\).

2. **Divide the Leading Terms:**

   The first term in the numerator is \(3x^2\) and the first term in the denominator is \(3x\). Divide these:

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

   So, the first term in the quotient is \(x\).

3. **Multiply and Subtract:**

   Multiply \(x\) by \(3x + 1\):

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

   Subtract this from the original numerator:

   \[
   (3x^2 - 11x - 4) - (3x^2 + x) = -12x - 4
   \]

4. **Repeat the Process:**

   Now, divide the new leading term \(-12x\) by \(3x\):

   \[
   \frac{-12x}{3x} = -4
   \]

   So, the next term in the quotient is \(-4\).

5. **Multiply and Subtract Again:**

   Multiply \(-4\) by \(3x + 1\):

   \[
   -4(3x + 1) = -12x - 4
   \]

   Subtract this from the current dividend:

   \[
   (-12x - 4) - (-12x - 4) = 0
   \]

   The remainder is \(0\).

6. **Conclusion:**

   The quotient after dividing is:

   \[
   x - 4
   \]

### Final Answer:

\[
\frac{3x^2 - 11x - 4}{3x + 1} = x - 4
\]

Would you like a more detailed breakdown, or do you have any questions?

### Related Questions:

1. How do you perform polynomial division when the remainder is not zero?
2. What are some practical applications of polynomial division?
3. Can this problem be solved using synthetic division?
4. How does the degree of the polynomials affect the division process?
5. How do you handle division when the divisor is a quadratic polynomial?

**Tip:** When performing polynomial division, always align terms according to their degrees to avoid mistakes during subtraction.

Learn how to simplify the expression (3x² - 11x - 4) / (3x + 1) using polynomial division. This detailed step-by-step solution covers key algebraic concepts like polynomial division, the division algorithm, and the remainder theorem. Suitable for high school students in Grades 9-12.

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