Question

1. the diagram represents a flower border that is 3 feet wide surrounding a rectangular sitting area. write an expression in factored form that represents the area of the flower border.

Explanation:

Step1: Find the area of the outer rectangle

The length of the outer rectangle is \(x + 3+3=x + 6\), and the width of the outer rectangle is \(x+3 + 3=x + 6\). The area of a rectangle is \(A=\text{length}\times\text{width}\), so the area of the outer rectangle \(A_{outer}=(x + 6)(x + 6)=(x + 6)^2\).

Step2: Find the area of the inner rectangle

The length and width of the inner rectangle are \(x\). The area of the inner rectangle \(A_{inner}=x\times x=x^{2}\).

Step3: Calculate the area of the flower - border

The area of the flower - border \(A = A_{outer}-A_{inner}\). Substitute the expressions for \(A_{outer}\) and \(A_{inner}\):

$$
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Another way: Using the formula \(a^{2}-b^{2}=(a + b)(a - b)\), where \(a=x + 6\) and \(b = x\). Then \(A=(x + 6+x)(x + 6-x)=(2x + 6)\times6=12(x + 3)\)

Answer:

\(12(x + 3)\)