x2 + y2 + 6x + 4y = 3

Get the x-terms next to each other.
Get the y-terms next to each other.
Skip a space after the x terms and after the y terms:

x2 + 6x     + y2 + 4y     = 3 

Complete the square out to the side or on scratch paper.

1. Multiply the coefficient of x by 1/2.   
2. Square what you get.                    
3. Add that to the space after the x-terms on the left and also
   add it to the right side.

x2 + 6x + 9 + y2 + 4y     = 3 + 9 

1. Multiply the coefficient of y by 1/2.   
2. Square what you get.                    
3. Add that to the space after the y-terms on the left and also
   add it to the right side.

x2 + 6x + 9 + y2 + 4y + 4 = 3 + 9 + 4

Factor the first three terms on the left side.
Factor the last three terms on the left side.
Combine the numbers on the right side.

(x + 3)(x + 3) + (y + 2)(y + 2) = 16

Notice that they factored as the same factor twice, so write
them as squares:

(x + 3)2 + (y + 2)2 = 16

Compare that to what you should have memorized.

(x - h)2 + (y - k)2 = r2 where (h,k) is the center and r is the radius.

Set corresponding things equal to each other and solve

-h = +3        -k = +2       r2 = 16
 h = -3         k = -2        r = √16
                              r = 4

Substitute for all letters except x and y



Simplify:



Center = (h,k) = (-3,-2),  radius = r = 4



Edwin