To find the area of the circle with the given equation `2x^2 + 2y^2 + 4x + 4y = 0` first find the radius of the circle.
First, simplify the equation by dividing every term by 2.
`x^2 + y^2 + 2x + 2y = 0`
Rewrite into general form of a circle `x^2 + y^2 = r^2` where r represents the radius of the circle.
To rewrite the equation, we must complete the square to find the radius.
`(x^2 + 2x + c) + (y^2 + 2y + d) = 0 + c + d`
To find c and d, divide middle term by 2 then square.
`(x^2 + 2x + 1) + (y^2 + 2y + 1) = 0 + 1 + 1`
`(x + 1)^2 + (y+1)^2 = 2`
This indicates that `r^2 = 2` so `r =sqrt(2)`
The radius of the circle is `sqrt(2).`
To find the area of a circle with radius `sqrt(2)` use the formula for area.
`A =pi* (sqrt(2))^2`
`A = 2pi`
The area of a circle given the equation `2x^2 + 2y^2 + 4x + 4y = 0` is `2pi.`