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The expression you need to simplify is:

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

### Step-by-step solution:

1. **Distribute the denominator**: Break the expression into two separate terms:
   \[
   \frac{5z^4}{-2z} + \frac{9z^3}{-2z}
   \]
   
2. **Simplify each term**:
   - For the first term: \(\frac{5z^4}{-2z} = \frac{5}{-2}z^{4-1} = -\frac{5}{2}z^3\)
   - For the second term: \(\frac{9z^3}{-2z} = \frac{9}{-2}z^{3-1} = -\frac{9}{2}z^2\)

3. **Combine the results**:
   \[
   -\frac{5}{2}z^3 - \frac{9}{2}z^2
   \]

Thus, the simplified form of the expression is:

\[
-\frac{5}{2}z^3 - \frac{9}{2}z^2
\]

Do you have any further questions or want more details?

### Related Questions:
1. How do you simplify expressions involving polynomials?
2. What are the rules for dividing terms with exponents?
3. How does distributing a denominator across a polynomial work?
4. Can similar terms with different exponents be combined?
5. How can negative signs in fractions affect simplification?

### Tip:
Always subtract exponents when dividing terms with the same base!

Divide: (5z^4 + 9z^3) / (-2z). Simplify your answer as much as possible.

Learn how to simplify the division of the polynomial expression (5z^4 + 9z^3) by -2z. This step-by-step solution explains the rules of exponents and polynomial division, suitable for students in Grades 9-10.

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