Question

the goddards want to cover their backyard with new grass. their backyard has the dimensions 3x - 5 feet by 4x - 10 feet. they have a deck with an inground pool in the back. the deck has an area of x + 8 by x - 2 feet. the pool has an area 4x - 3 by 2x feet. what is the area of the region of the yard they want to cover with new grass?

Explanation:

Step1: Calculate the area of the backyard

The area of a rectangle is length times width. So the area of the backyard with dimensions \(3x - 5\) and \(4x - 10\) is \((3x - 5)(4x - 10)\).
Using the FOIL method:
\[

$$\begin{align*} (3x - 5)(4x - 10)&=3x\times4x+3x\times(- 10)-5\times4x+(-5)\times(-10)\\ &=12x^{2}-30x - 20x + 50\\ &=12x^{2}-50x + 50 \end{align*}$$

\]

Step2: Calculate the area of the deck

The deck has dimensions \(x + 8\) and \(x - 2\), so its area is \((x + 8)(x - 2)\).
Using the FOIL method:
\[

$$\begin{align*} (x + 8)(x - 2)&=x\times x+x\times(-2)+8\times x+8\times(-2)\\ &=x^{2}-2x + 8x-16\\ &=x^{2}+6x - 16 \end{align*}$$

\]

Step3: Calculate the area of the pool

The pool has dimensions \(4x - 3\) and \(2x\), so its area is \((4x - 3)(2x)\).
\[

$$\begin{align*} (4x - 3)(2x)&=4x\times2x-3\times2x\\ &=8x^{2}-6x \end{align*}$$

\]

Step4: Calculate the total area of the deck and the pool

Add the area of the deck and the area of the pool:
\[

$$\begin{align*} (x^{2}+6x - 16)+(8x^{2}-6x)&=x^{2}+8x^{2}+6x-6x - 16\\ &=9x^{2}-16 \end{align*}$$

\]

Step5: Calculate the area to be covered with grass

Subtract the total area of the deck and pool from the area of the backyard:
\[

$$\begin{align*} (12x^{2}-50x + 50)-(9x^{2}-16)&=12x^{2}-50x + 50 - 9x^{2}+16\\ &=(12x^{2}-9x^{2})-50x+(50 + 16)\\ &=3x^{2}-50x + 66 \end{align*}$$

\]

Answer:

The area of the region to be covered with new grass is \(3x^{2}-50x + 66\) square feet.