Select the correct answer. Sam is installing a walkway around a rectangular flower patch in his garden. The flower patch is 12 feet long and 6 feet wide. The width of the walkway is $$ x $$ feet. Sam created function $$ A $$ to represent the total area taken up by the flower patch and walkway by multiplying the functions modeling the new total length and width. A(x) $$ =4 x^{2}+36 x+72 $$ What does $$ 4 x^{2}+36 x $$ represent in this function? A. the total area of the flower patch B. the total area of the walkway C. the area of the walkway along the length of the flower patch D. the area of the walkway along the width of the flower patch Next

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To determine what $$ 4x^{2} + 36x $$ represents in the function $$ A(x) = 4x^{2} + 36x + 72 $$, let's break down the problem:

1. **Flower Patch Dimensions:**
   - Length: 12 feet
   - Width: 6 feet
   - **Area of Flower Patch:** $$ 12 \times 6 = 72 $$ square feet

2. **Walkway Dimensions:**
   - Width of Walkway: $$ x $$ feet
   - **Total Length Including Walkway:** $$ 12 + 2x $$ feet
   - **Total Width Including Walkway:** $$ 6 + 2x $$ feet
   - **Total Area (Flower Patch + Walkway):** 
     $$
     (12 + 2x)(6 + 2x) = 72 + 24x + 12x + 4x^{2} = 72 + 36x + 4x^{2}
     $$
     This matches the given function $$ A(x) = 4x^{2} + 36x + 72 $$.

3. **Interpreting $$ 4x^{2} + 36x $$:**
   - Subtracting the area of the flower patch from the total area:
     $$
     A(x) - 72 = 4x^{2} + 36x
     $$
   - Therefore, $$ 4x^{2} + 36x $$ represents the **total area of the walkway** around the flower patch.

**Correct Answer:**

**B. the total area of the walkway**
$$ 4x^{2} + 36x $$ represents the total area of the walkway.

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