write the standard form of the equation of the circle.

Explanation:

Step1: Identify circle center

The center $(h,k)$ is $(-4, -4)$.

Step2: Calculate radius length

Use distance formula on center $(-4,-4)$ and point $(0,0)$ on the circle: $\text{Radius } r = \sqrt{(-4-0)^2 + (-4-0)^2} = \sqrt{16+16} = \sqrt{32}$

Step3: Plug into standard circle formula

Standard form: $(x-h)^2 + (y-k)^2 = r^2$ Substitute $h=-4$, $k=-4$, $r^2=32$: $(x-(-4))^2 + (y-(-4))^2 = 32$ Simplify signs: $(x+4)^2 + (y+4)^2 = 32$

Answer:

$\boldsymbol{(x+4)^2 + (y+4)^2 = 32}$