We want to convert into standard form, the equation
x² + y² + 12x + 2y - 1 = 0
Note that
(x + a)² = x² + 2ax + a²
so that
x² + 2ax = (x + a)² - a²
Therefore
x² + y² + 12x + 2y - 1 = 0
x² + 12x + y² + 2y = 1
(x + 6)² - 6² + (y + 1)² - 1² = 1
(x + 6)² + (y + 1)² - 37 = 1
(x + 6)² + (y + 1)² = 38
Theis equation represents a circle with center at (-6, -1) and radius √(38), as shown below.
Answer: (x + 6)² + (y + 1)² = 38
