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Let's work through this division step-by-step.

### Problem:
\[
\frac{-8y^2 x^4 + 5y^6 x^4}{-2y^4 x^2}
\]

### Step 1: Break down the terms
We can split the expression as follows:

\[
\frac{-8y^2 x^4}{-2y^4 x^2} + \frac{5y^6 x^4}{-2y^4 x^2}
\]

### Step 2: Simplify each term

1. **First Term:**
   \[
   \frac{-8y^2 x^4}{-2y^4 x^2} = \frac{-8}{-2} \cdot \frac{y^2}{y^4} \cdot \frac{x^4}{x^2}
   \]
   Simplifying each part:
   \[
   \frac{-8}{-2} = 4, \quad \frac{y^2}{y^4} = \frac{1}{y^2}, \quad \frac{x^4}{x^2} = x^2
   \]
   So the first term becomes:
   \[
   4 \cdot \frac{1}{y^2} \cdot x^2 = \frac{4x^2}{y^2}
   \]

2. **Second Term:**
   \[
   \frac{5y^6 x^4}{-2y^4 x^2} = \frac{5}{-2} \cdot \frac{y^6}{y^4} \cdot \frac{x^4}{x^2}
   \]
   Simplifying each part:
   \[
   \frac{5}{-2} = -\frac{5}{2}, \quad \frac{y^6}{y^4} = y^2, \quad \frac{x^4}{x^2} = x^2
   \]
   So the second term becomes:
   \[
   -\frac{5}{2} \cdot y^2 \cdot x^2 = -\frac{5x^2 y^2}{2}
   \]

### Step 3: Combine the simplified terms
Now, combine the two simplified terms:
\[
\frac{4x^2}{y^2} - \frac{5x^2 y^2}{2}
\]

This is the simplified expression. If you have any further questions, let me know!

### Follow-up questions:
1. How do we further simplify rational expressions like this?
2. What properties of exponents help in this simplification?
3. How does division of polynomials work?
4. What are some common mistakes when simplifying fractions?
5. Can we factor these expressions for further simplification?

**Tip:** Always remember to simplify individual terms separately before combining them.

Divide: \[\frac{-8y^2 x^4 + 5y^6 x^4}{-2y^4 x^2}\]. Simplify your answer as much as possible.

Learn how to simplify a complex rational expression involving variables and exponents, \(\frac{-8y^2 x^4 + 5y^6 x^4}{-2y^4 x^2}\), using exponent rules and algebraic techniques. This step-by-step solution is perfect for high school students in Grades 10-12.

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