The equation does not represent a circle because the radius ≤ 0.
x² + y² + 13 = 4x + 6y
x² - 4x + y² - 6y + 13 = 0
x² - 4x - (4/2)² + y² - 6y - (6/2)² = -13
x² - 4x + y² - 6y - 4 - 9 = -13
x² - 4x + y² - 6y - 13 = - 13
x² - 4x + y² - 6y = -13 + 13
x² - 4x + y² - 6y = 0
or:
(x-2)² + (y-3)² = -13