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Let's solve the expression step by step.

The given expression is:

\[
\frac{-24x^6y^7 + 8x^6y}{4x^4y^5}
\]

### Step 1: Separate the Terms
We can separate the fraction into two terms:

\[
\frac{-24x^6y^7}{4x^4y^5} + \frac{8x^6y}{4x^4y^5}
\]

### Step 2: Simplify Each Term Individually

1. **Simplify the first term** \(\frac{-24x^6y^7}{4x^4y^5}\):

   - Divide the coefficients: \(\frac{-24}{4} = -6\).
   - For the \(x\) terms: \(x^6 / x^4 = x^{6-4} = x^2\).
   - For the \(y\) terms: \(y^7 / y^5 = y^{7-5} = y^2\).

   Thus, the first term simplifies to: \(-6x^2y^2\).

2. **Simplify the second term** \(\frac{8x^6y}{4x^4y^5}\):

   - Divide the coefficients: \(\frac{8}{4} = 2\).
   - For the \(x\) terms: \(x^6 / x^4 = x^{6-4} = x^2\).
   - For the \(y\) terms: \(y / y^5 = y^{1-5} = y^{-4}\).

   Thus, the second term simplifies to: \(2x^2y^{-4}\).

### Step 3: Combine the Simplified Terms

The simplified expression becomes:

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

### Step 4: Factor Out the Common Terms

Notice that \(2x^2\) is a common factor:

\[
2x^2(-3y^2 + y^{-4})
\]

### Final Answer

\[
\boxed{2x^2(-3y^2 + y^{-4})}
\]

If you prefer, this can also be left in its expanded form: \(-6x^2y^2 + 2x^2y^{-4}\).

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Would you like any further explanation, or do you have any questions?

Here are some related questions for further practice:

1. How do you divide terms with the same base when they have exponents?
2. What happens when you have a negative exponent, and how can you rewrite it?
3. Can you factor out terms that are common in more complex expressions?
4. How would you simplify an expression with three terms instead of two?
5. What happens if you multiply an expression like this with another polynomial?

**Tip:** When dividing terms with exponents, always remember to subtract the exponents when the bases are the same!

Divide and simplify: (-24x^6y^7 + 8x^6y) / (4x^4y^5)

Learn how to simplify the algebraic expression (-24x^6y^7 + 8x^6y) / (4x^4y^5) using exponent rules. This problem demonstrates key concepts like simplifying terms with exponents and factoring. Suitable for students in Grades 8-10.

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