To divide the expression \((- \frac{5}{6} u^2 v^2 + \frac{1}{4} u^2 v^2)\) by \(6u^2v^2\), we proceed step by step:

### Step 1: Factor out common terms
First, notice that both terms in the numerator contain \(u^2 v^2\), so we can factor that out:
\[
- \frac{5}{6} u^2 v^2 + \frac{1}{4} u^2 v^2 = u^2 v^2 \left(- \frac{5}{6} + \frac{1}{4}\right)
\]
Now, the expression becomes:
\[
\frac{u^2 v^2 \left(- \frac{5}{6} + \frac{1}{4}\right)}{6 u^2 v^2}
\]

### Step 2: Simplify
The \(u^2 v^2\) terms in the numerator and denominator cancel out, leaving us with:
\[
\frac{- \frac{5}{6} + \frac{1}{4}}{6}
\]

### Step 3: Combine the fractions
Now we need to combine \(- \frac{5}{6}\) and \(\frac{1}{4}\). To do this, find a common denominator:
\[
\text{LCD of } 6 \text{ and } 4 = 12
\]
Rewrite the fractions:
\[
- \frac{5}{6} = - \frac{10}{12}, \quad \frac{1}{4} = \frac{3}{12}
\]
Now, combine them:
\[
- \frac{10}{12} + \frac{3}{12} = - \frac{7}{12}
\]

### Step 4: Divide by 6
Now we need to divide \(- \frac{7}{12}\) by 6. Dividing by 6 is the same as multiplying by \(\frac{1}{6}\):
\[
- \frac{7}{12} \times \frac{1}{6} = - \frac{7}{72}
\]

### Final Answer:
\[
\frac{- \frac{5}{6} u^2 v^2 + \frac{1}{4} u^2 v^2}{6 u^2 v^2} = - \frac{7}{72}
\]

Would you like more details or have any questions?

Here are five related questions for further exploration:
1. How would the result change if the denominator was \(12u^2v^2\)?
2. Can you apply this process to expressions involving different variables and powers?
3. What if the numerator had additional terms, like a third fraction?
4. How does the division change if negative exponents are involved?
5. How do you divide polynomials in general, when factoring is not possible?

**Tip:** Always look for common factors to simplify before dividing complex expressions.

Divide (-5/6u^2v^2 + 1/4u^2v^2) by 6u^2v^2.

Learn how to divide the rational expression (-5/6u^2v^2 + 1/4u^2v^2) by 6u^2v^2 with a clear, step-by-step explanation. This problem involves algebraic simplification, factoring, and fraction division. Perfect for students in Grades 9-12.

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