Question

the miller family plans to install carpet in their game room. the room is the shape of a square with side length ( w ) meters. the entire room will be carpeted except for a small square area with side length 2 meters near the door where they will not install carpet. find an expression for the area that will be carpeted. write the expression in factored form.

Explanation:

Step1: Find area of game room

The game room is a square with side length \( w \) meters. The area of a square is side length squared, so the area of the game room is \( w^2 \) square meters.

Step2: Find area of non - carpeted region

The non - carpeted region is a square with side length 2 meters. The area of this square is \( 2\times2 = 2^2=4 \) square meters.

Step3: Find area of carpeted region

The area to be carpeted is the area of the game room minus the area of the non - carpeted region. So, the area \( A=w^{2}-4 \).
We can factor \( w^{2}-4 \) using the difference of squares formula \( a^{2}-b^{2}=(a + b)(a - b) \), where \( a = w \) and \( b = 2 \). So \( w^{2}-4=(w + 2)(w - 2) \).

Answer:

The expression for the area that will be carpeted in factored form is \((w + 2)(w - 2)\) square meters.