Question

A family is building a rectangular patio in their backyard. The rectangular yard has dimensions of (8x+2) by (6x + 3) and they are planning the patio to be (x + 5) by (3x + 1). What is the area of the remaining yard after the patio has been built?

(1 point)
3x 2 + 16x + 5
48x 2 + 36x + 6
51x 2 + 52x + 11
45x 2 + 20x + 1

Answer 1

So, if you multiply each polynomial by eachother, the area of the patio is 3x^2+16x+5, and the area of the yard is 48x^2+36x+6. If you subtract patio from the yard, the remaining value is 45x^2+20x+1, which is the remaining area of the yard, so the correct answer is D

Answer 2

Option 4th is correct

`45x^2+20x+1`

Step-by-step explanation:

Area of rectangle(A) is given by:

`A = lw`

where,

l is the length and w is the width of the rectangle

As per the statement:

. The rectangular yard has dimensions of (8x+2) by (6x + 3)

⇒
`\text{Area of yard} = (8x+2) \cdot (6x+3)`

⇒
`\text{Area of yard} = 8x(6x+3)+2(6x+3)`

Using the distributive property:
`a \cdot (b+c)=a\cdot b+ a\cdot c`

⇒
`\text{Area of yard} =48x^2+24x+12x+6 = 48x^2+36x+6`

It is also given that:

They are planning the patio to be (x + 5) by (3x + 1).

`\text{Area of patio} = (x+5) \cdot (3x+1) =3x^2+x+15x+5 = 3x^2+16x+5`

We have to find the the area of the remaining yard after the patio has been built

`\text{Remaining area of yard} = \text{Area of yard} -\text{Area of the patio built}`

⇒
`\text{Remaining area of yard} =48x^2+36x+6-(3x^2+16x+5)` ⇒
`\text{Remaining area of yard} =48x^2+36x+6-3x^2-16x-5`

Combine like term;

⇒
`\text{Remaining area of yard} =45x^2+20x+1`

Therefore, the area of the remaining yard after the patio has been built is,
`45x^2+20x+1`