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The expression provided is:

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

To divide these polynomials, we can start by rearranging the terms of the numerator in standard form (from highest to lowest degree):

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

### Step 1: Polynomial Division

We'll use long division to divide the polynomials.

1. Divide the first term of the numerator \(-9x^4\) by the first term of the denominator \(-3x^2\):
   \[
   \frac{-9x^4}{-3x^2} = 3x^2
   \]
   
2. Multiply the entire divisor \((-3x^2 - 4)\) by \(3x^2\) and subtract from the original numerator:
   \[
   (3x^2)(-3x^2 - 4) = -9x^4 - 12x^2
   \]
   Subtracting:
   \[
   (-9x^4 + 15x^3 - 18x^2 + 11x - 4) - (-9x^4 - 12x^2) = 15x^3 - 6x^2 + 11x - 4
   \]

3. Repeat the process with the new polynomial \(15x^3 - 6x^2 + 11x - 4\).

   Divide the first term \(15x^3\) by the first term of the divisor \(-3x^2\):
   \[
   \frac{15x^3}{-3x^2} = -5x
   \]

4. Multiply \(-5x\) by the divisor:
   \[
   (-5x)(-3x^2 - 4) = 15x^3 + 20x
   \]
   Subtracting:
   \[
   (15x^3 - 6x^2 + 11x - 4) - (15x^3 + 20x) = -6x^2 - 9x - 4
   \]

5. Repeat one more time:

   Divide \(-6x^2\) by \(-3x^2\):
   \[
   \frac{-6x^2}{-3x^2} = 2
   \]

6. Multiply \(2\) by the divisor:
   \[
   (2)(-3x^2 - 4) = -6x^2 - 8
   \]
   Subtracting:
   \[
   (-6x^2 - 9x - 4) - (-6x^2 - 8) = -9x + 4
   \]

The quotient is \(3x^2 - 5x + 2\) and the remainder is \(-9x + 4\).

Thus, the complete expression is:

\[
3x^2 - 5x + 2 + \frac{-9x + 4}{-3x^2 - 4}
\]

### Summary

The result of the division is:

\[
3x^2 - 5x + 2 + \frac{-9x + 4}{-3x^2 - 4}
\]

Would you like to go over any specific steps in more detail?

### Follow-up Questions

1. How does polynomial long division differ from regular long division?
2. What are the applications of polynomial division in calculus?
3. Can synthetic division be used for this problem? Why or why not?
4. What is the significance of the remainder in polynomial division?
5. How can polynomial division be applied in finding asymptotes of rational functions?

### Tip

When performing polynomial division, keep your work organized by aligning similar terms, as this reduces the risk of calculation errors.

(15x^3 - 18x^2 - 9x^4 - 4 + 11x) ÷ (-3x^2 - 4)

Learn how to divide the polynomial (15x^3 - 18x^2 - 9x^4 + 11x - 4) by (-3x^2 - 4) using long division. This detailed step-by-step solution covers key algebraic concepts such as polynomial division, remainders, and simplifying expressions.

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