First, the asymptotes. An easy way to create an asymptote is to create an expression in which the denominator of a fraction becomes zero at one particular value of the variable. So an asymptote at x=4 (vertical) is created by x-4 in a denominator, and an asymptote at y=-2 (horizontal) is created by y+2 in a denominator. Therefore, if we have y+2=1/(x-4), which is the same as x-4=1/(y+2) we have an equation with the required asymptotes.
Now the point singularity at (3,-6). This is created by generating the incalculable zero divided by zero. So we need the fraction (x-3)/(y+6) or (y+6)/(x-3) to create the singularity (hole). This must be combined with the asymptote equation.
One possibility is y+2=1/(x-4)+(y+6)/(x-3). In this form, the singularity at (3,-6) is clearly shown.
From this y=(21x-2x²-51)/(x-4)², but this does not contain the singularity, it only demonstrates the asymptotes.
There are other possibilities.
The curve is shown in red and the asymptotes in green.