If the denominator is a factor of the numerator there is no OA.
For example: y=(x²-1)/(x-1), y=(x²+3x)/x. These functions have a “hole” but no OA.
IN GENERAL
For a quadratic numerator:
y=(ax²+bx-bc-ac²)/(x-c), where a, b and c are arbitrary constants and a≠0.