Let's break down the given problem and analyze what $4x^2 + 36x$ represents in the function $A(x) = 4x^2 + 36x + 72$.

The flower patch is a rectangle with dimensions 12 feet by 6 feet. The walkway width is $x$ feet, and Sam is calculating the total area of the flower patch and the walkway.

1. Area of the flower patch (without the walkway):
   The area of the flower patch is $12 \times 6 = 72$ square feet. This is the constant term, $72$, in the equation $A(x)$.

2. Total dimensions with the walkway:
   After adding the walkway of width $x$ around the patch, the new dimensions are:

   • Length: $12 + 2x$
   • Width: $6 + 2x$

3. Total area (patch + walkway):
   The total area with the walkway would be:
   $A(x) = (12 + 2x)(6 + 2x)$ Expanding this expression: $A(x) = 72 + 24x + 12x + 4x^2 = 4x^2 + 36x + 72$ This matches the given function $A(x) = 4x^2 + 36x + 72$.

Now, the question asks what $4x^2 + 36x$ represents. This part of the equation comes from the expanded terms that include $x$, which is the additional area contributed by the walkway around the flower patch.

• $4x^2$: This represents the area of the corners added by the walkway.
• $36x$: This represents the area of the walkway along the sides of the flower patch.

Thus, $4x^2 + 36x$ represents the total area of the walkway itself. The correct answer is:

D. The total area of the walkway.

Would you like further details on this, or do you have any questions?

Here are some follow-up questions related to this problem:

1. How would the function change if the dimensions of the flower patch were different?
2. What if the walkway only surrounded three sides of the flower patch—how would that affect the function?
3. How can you derive the function from just the description, without expanding?
4. How would you determine the width $x$ of the walkway if you knew the total area?
5. Can this approach be generalized to other shapes like circles?

Tip: When expanding binomials in word problems, visualize how each term corresponds to parts of the geometric figure!