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The given division problem is:

\[
\frac{-9x^4 + 15x^3 - 18x^2 + 11x - 4}{-3x^2 - 4}
\]

### Steps to solve using long division:
We need to divide the numerator \(-9x^4 + 15x^3 - 18x^2 + 11x - 4\) by the denominator \(-3x^2 - 4\).

1. **First Division Step**:
   - Divide the leading term of the numerator \(-9x^4\) by the leading term of the denominator \(-3x^2\):
     \[
     \frac{-9x^4}{-3x^2} = 3x^2
     \]
   - Multiply \((3x^2)\) by \((-3x^2 - 4)\):
     \[
     3x^2 \cdot (-3x^2 - 4) = -9x^4 - 12x^2
     \]
   - Subtract this result from the numerator:
     \[
     (-9x^4 + 15x^3 - 18x^2) - (-9x^4 - 12x^2) = 15x^3 - 6x^2
     \]

2. **Second Division Step**:
   - Divide the new leading term \(15x^3\) by \(-3x^2\):
     \[
     \frac{15x^3}{-3x^2} = -5x
     \]
   - Multiply \((-5x)\) by \((-3x^2 - 4)\):
     \[
     -5x \cdot (-3x^2 - 4) = 15x^3 + 20x
     \]
   - Subtract this from the new polynomial:
     \[
     (15x^3 - 6x^2 + 11x) - (15x^3 + 20x) = -6x^2 - 9x
     \]

3. **Third Division Step**:
   - Divide the new leading term \(-6x^2\) by \(-3x^2\):
     \[
     \frac{-6x^2}{-3x^2} = 2
     \]
   - Multiply \((2)\) by \((-3x^2 - 4)\):
     \[
     2 \cdot (-3x^2 - 4) = -6x^2 - 8
     \]
   - Subtract this:
     \[
     (-6x^2 - 9x - 4) - (-6x^2 - 8) = -9x + 4
     \]

4. **Conclusion**:
   Now, the remainder is \(-9x + 4\). So, the quotient is:
   \[
   3x^2 - 5x + 2
   \]
   and the remainder is:
   \[
   \frac{-9x + 4}{-3x^2 - 4}
   \]

### Final Answer:
\[
3x^2 - 5x + 2 + \frac{-9x + 4}{-3x^2 - 4}
\]

Would you like further clarification or details?

Here are five related questions:
1. How do you perform polynomial long division for higher-order polynomials?
2. What is the significance of the remainder in polynomial division?
3. Can you explain the role of synthetic division in solving polynomial division problems?
4. What happens if the remainder is zero in polynomial division?
5. How is the quotient used when simplifying rational expressions?

**Tip**: Always align terms by their degree when performing polynomial long division to avoid errors.

Divide (-9x^4 + 15x^3 - 18x^2 + 11x - 4) ÷ (-3x^2 - 4) and express the answer in the form Quotient + Remainder / (-3x^2 - 4).

Learn how to divide the polynomial (-9x^4 + 15x^3 - 18x^2 + 11x - 4) by (-3x^2 - 4) using polynomial long division. The solution involves finding the quotient and the remainder, which is expressed as a fraction. Suitable for students in Grades 10-12.

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