I have no idea if this is the right place to post this, but here goes:
x^2 - 4x + 8y + 4 = 0
If someone could please detail each individual step, that would be awesome.
You don't have to complete the square if you move things around:
$\displaystyle x^2-4x+8y+4=0$
And let's switch the places of the $\displaystyle 8y$ and the $\displaystyle 4$
$\displaystyle x^2-4x+4+8y=0$
Now let's factor
$\displaystyle (x-2)(x-2)+8y=0$
$\displaystyle (x-2)^2+8y=0$
Well we have $\displaystyle (x-2)^2+8y=0$
And you want the x on the other side so lets subtract $\displaystyle (x-2)^2$ from both sides
$\displaystyle (x-2)^2+8y-(x-2)^2=-(x-2)^2$
$\displaystyle 8y=-(x-2)^2$
And divide by 8 to get
$\displaystyle y=-\frac{1}{8}(x-2)^2$
In this case $\displaystyle k=0$