Answer:
Option 4th is correct
\(45x^2+20x+1\)
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, \(45x^2+20x+1\)
