Given the problem:

1. The side of the square field is $x + 2y$, so its area is the square of the side:

   = x^2 + 4xy + 4y^2$$

2. The radius of the circular path is $x + y$, so the area of the circular path is: $A_{\text{circle}} = \pi (x + y)^2 = \pi (x^2 + 2xy + y^2)$

3. The remaining area of the field (the area of the square minus the area of the circle) is: $A_{\text{remaining}} = A_{\text{square}} - A_{\text{circle}} = (x^2 + 4xy + 4y^2) - \pi(x^2 + 2xy + y^2)$ Simplifying: $A_{\text{remaining}} = (x^2 + 4xy + 4y^2) - \pi(x^2 + 2xy + y^2)$ $A_{\text{remaining}} = (1 - \pi)x^2 + (4 - 2\pi)xy + (4 - \pi)y^2$

Now, to find the cost of leveling the remaining land:

The cost is given by the rate $x^2 + 2x - 4$, so the total cost for the remaining land is: $\text{Cost} = \left( x^2 + 2x - 4 \right) \times A_{\text{remaining}}$ Substituting the expression for $A_{\text{remaining}}$: $\text{Cost} = (x^2 + 2x - 4) \times \left( (1 - \pi)x^2 + (4 - 2\pi)xy + (4 - \pi)y^2 \right)$

This expression represents the total cost of leveling the land.

Would you like further details on the cost expansion, or help with any specific part of the problem?

Follow-up questions:

1. What happens if the rate of leveling the land changes?
2. How does the cost change if $x$ and $y$ are equal?
3. What is the area when the square field has side 10 and the circular path has radius 7?
4. Can we calculate the cost for a specific value of $x$ and $y$?
5. How would the calculation differ for a non-square field?

Tip:

When solving area problems, visualize the geometry to better understand the relationships between shapes.