<s>[INST] Complete the squares to find the center and radius of the circle I've had trouble on this one problem for a couple days.
Complete the square on the X and Y terms to find the center and radius of the circle.
 $x^2+2x+y^2-4y=-4\:\:$
 [/INST] The equation of the circle with center at $(h,k)$ and radius $a$ is given by this form,
$$(x-h)^2+(y-k)^2=a^2$$
If you look at the your equation, you will see that it can be expressed in the above form, your equation is addition of two squares as shown below:
$(a+b)^2=\underbrace{(\text{first term})^2}+\underbrace{2\times(\text{first term})(\text{second term})}+\underbrace{(\text{second term})^2}$we have$\hspace{40 pt}x^2 \hspace{80 pt}2.x.1 \hspace{80 pt}\text{missing}$completing square is adding the $\color{red}{\text{square of half of coefficeint of the middle term}}$so that we can express an expression as a square of some expression
$(x+1)^2=x^2+2x+\color{red}1\Rightarrow x^2+2x=(x+1)^2-1...(I)$ $(y-2)^2=y^2-4y+\color{red}4\Rightarrow y^2-4y=(y-2)^2-4...(II)$
Add $(I)$ and $(II)$
$x^2+2x+y^2-4y=(x+1)^2-1+(y-2)^2-4$
$\therefore (x+1)^2-1+(y-2)^2-4=-4$
$\therefore(x+1)^2+(y-2)^2=1$,
$\therefore(x-(-1))^2+(y-2)^2=1$ 
compare this with the first equation of the circle and we get radius$=a=1, \text{center=} (h,k)=(-1,2)$
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