Answer:
Option 4th is correct
[tex]45x^2+20x+1[/tex]
Explanation:
Area of rectangle(A) is given by:
[tex]A = lw[/tex]
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)
⇒[tex]\text{Area of yard} = (8x+2) \cdot (6x+3)[/tex]
⇒[tex]\text{Area of yard} = 8x(6x+3)+2(6x+3)[/tex]
Using the distributive property: [tex]a \cdot (b+c)=a\cdot b+ a\cdot c[/tex]
⇒[tex]\text{Area of yard} =48x^2+24x+12x+6 = 48x^2+36x+6[/tex]
It is also given that:
They are planning the patio to be (x + 5) by (3x + 1).
[tex]\text{Area of patio} = (x+5) \cdot (3x+1) =3x^2+x+15x+5 = 3x^2+16x+5[/tex]
We have to find the the area of the remaining yard after the patio has been built
[tex]\text{Remaining area of yard} = \text{Area of yard} -\text{Area of the patio built}[/tex]
⇒[tex]\text{Remaining area of yard} =48x^2+36x+6-(3x^2+16x+5)[/tex] ⇒[tex]\text{Remaining area of yard} =48x^2+36x+6-3x^2-16x-5[/tex]
Combine like term;
⇒[tex]\text{Remaining area of yard} =45x^2+20x+1[/tex]
Therefore, the area of the remaining yard after the patio has been built is, [tex]45x^2+20x+1[/tex]
