\frac{\frac{m^2-mn}{6m^2+11mn+3n^2}\div \frac{m^2-n^2}{2m^2-mn-6n^2}}{\frac{4m^2-7mn-2n^2}{3m^2+7mn+2n^2^'}}

The given expression is:

\[
\frac{\frac{m^2 - mn}{6m^2 + 11mn + 3n^2} \div \frac{m^2 - n^2}{2m^2 - mn - 6n^2}}{\frac{4m^2 - 7mn - 2n^2}{3m^2 + 7mn + 2n^2}}
\]

### Step 1: Simplify the division in the numerator
When dividing by a fraction, we multiply by its reciprocal. So, simplify the numerator:

\[
\frac{m^2 - mn}{6m^2 + 11mn + 3n^2} \div \frac{m^2 - n^2}{2m^2 - mn - 6n^2}
\]

can be rewritten as:

\[
\frac{m^2 - mn}{6m^2 + 11mn + 3n^2} \times \frac{2m^2 - mn - 6n^2}{m^2 - n^2}
\]

Now, this is:

\[
\frac{(m^2 - mn)(2m^2 - mn - 6n^2)}{(6m^2 + 11mn + 3n^2)(m^2 - n^2)}
\]

### Step 2: Combine the full expression
Now, we substitute this back into the original expression:

\[
\frac{\frac{(m^2 - mn)(2m^2 - mn - 6n^2)}{(6m^2 + 11mn + 3n^2)(m^2 - n^2)}}{\frac{4m^2 - 7mn - 2n^2}{3m^2 + 7mn + 2n^2}}
\]

This is equivalent to multiplying by the reciprocal of the denominator:

\[
\frac{(m^2 - mn)(2m^2 - mn - 6n^2)}{(6m^2 + 11mn + 3n^2)(m^2 - n^2)} \times \frac{3m^2 + 7mn + 2n^2}{4m^2 - 7mn - 2n^2}
\]

### Step 3: Final expression
So, the fully simplified expression is:

\[
\frac{(m^2 - mn)(2m^2 - mn - 6n^2)(3m^2 + 7mn + 2n^2)}{(6m^2 + 11mn + 3n^2)(m^2 - n^2)(4m^2 - 7mn - 2n^2)}
\]

This is the most simplified version without further factoring (if possible) or solving for specific values of \(m\) and \(n\).

Would you like further details or have any questions?

Here are five related questions:
1. How can we factor expressions involving quadratic terms?
2. How do we simplify complex rational expressions?
3. What is the difference between multiplying and dividing rational expressions?
4. How does multiplying by the reciprocal simplify division of fractions?
5. Can specific values of \(m\) and \(n\) lead to further simplifications?

**Tip:** When dividing fractions, always remember to multiply by the reciprocal of the second fraction for easier simplification.

The query submitted by the user: \frac{\frac{m^2-mn}{6m^2+11mn+3n^2}\div \frac{m^2-n^2}{2m^2-mn-6n^2}}{\frac{4m^2-7mn-2n^2}{3m^2+7mn+2n^2}}

Learn how to simplify a complex rational expression involving multiple fractions and factoring. This step-by-step solution explores key concepts such as factoring quadratics and using the reciprocal property to simplify rational expressions. Ideal for students in Grades 10-12.

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