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The problem you shared contains an application exercise where the area of a lot is given by the expression \( 6m^3n - 12m^2n^2 + 5mn + 4mn \), and the width is \( 3m^2n \). We are asked to find the expression that represents the length of the lot.

### Step-by-Step Solution:

The formula for the area of a rectangle is:
\[
\text{Area} = \text{Length} \times \text{Width}
\]
Given:
- Area = \( 6m^3n - 12m^2n^2 + 5mn + 4mn \)
- Width = \( 3m^2n \)

1. **Simplify the area expression:**
   Combine like terms:
   \[
   5mn + 4mn = 9mn
   \]
   So, the area expression becomes:
   \[
   6m^3n - 12m^2n^2 + 9mn
   \]

2. **Solve for the length:**
   To find the length, divide the area by the width:
   \[
   \text{Length} = \frac{\text{Area}}{\text{Width}} = \frac{6m^3n - 12m^2n^2 + 9mn}{3m^2n}
   \]

   Now, divide each term in the area expression by \( 3m^2n \):
   \[
   \frac{6m^3n}{3m^2n} = 2m
   \]
   \[
   \frac{-12m^2n^2}{3m^2n} = -4n
   \]
   \[
   \frac{9mn}{3m^2n} = \frac{9}{3m} = \frac{3}{m}
   \]

   So, the expression for the length is:
   \[
   \text{Length} = 2m - 4n + \frac{3}{m}
   \]

### Final Answer:
The expression that represents the length of the lot is:
\[
\boxed{2m - 4n + \frac{3}{m}}
\]

Would you like further clarification or more details on any of the steps?

### Related Questions:
1. How do you find the width if the area and length are known?
2. How can you simplify polynomial expressions involving multiple terms?
3. How would the solution change if the width was expressed as a different term?
4. What techniques are useful for factoring expressions like the one given?
5. How would the answer change if one of the variables was given a specific value?

### Tip:
When dividing polynomials, always check for common factors and simplify each term carefully.

If a lot has an area represented by the expression 6m^3n - 12m^2n^2 + 5mn + 4mn, and its width is 3m^2n, find the expression that corresponds to its length.

Learn how to find the length of a lot when the area is given by the polynomial expression 6m^3n - 12m^2n^2 + 5mn + 4mn and the width is 3m^2n. This problem covers key algebraic concepts, such as polynomial division and simplifying expressions. Suitable for high school students in Grades 10-12.

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