Let's try to complete the squares.
The x-part starts with [tex]x^2+10x[/tex], which is the beginning of [tex]x^2+10x+25=(x+5)^2[/tex]. So, we'll think of [tex]x^2+10x[/tex] as [tex](x+5)^2-25[/tex]
Similarly, we have that
[tex]y^2+12y = (y+6)^2-36[/tex]
So, the equation becomes
[tex]x^2 + y^2 + 10x + 12y + 25 = 0 \iff (x+5)^2-25 + (y+6)^2-36+25=0 \iff (x+5)^2+ (y+6)^2-36=0 \iff (x+5)^2+ (y+6)^2=36[/tex]
Now we have writte the equation of the circle in the form
[tex](x-k)^2+(y-h)^2=r^2[/tex]
When the equation is in this form, everything is more simple: the center is [tex](k,h)[/tex] and the radius is [tex]r[/tex].
