x2 + 6x + y2 = 4
Complete the square for x: ( x2 + 6x + 9 ) + y2 = 4 + 9
Factor: (x + 3)2 + y2 = 13
The problem has now become: what two perfect squares have a sum of 13?
-- they are 4 and 9.
So, either (x + 3)2 = 4 and y2 = 9
---> x + 3 = 2 y = 3
x = -1 y = -3
or x + 3 = -2
x = -5
Combining these answers to get: (-1,3), (-1,-3), (-5,3), (-5,-3)
Or: (x + 3)2 = 9 and y2 = 4
---> x + 3 = 3 y = 2
x = 0 y = -2
or x + 3 = -3
x = -6
Combining these answers to get: (0,2), (0,-2), (-6,2), (-6,-2)