The other tutor has a sign wrong.

9x² - 36x - 4y² - 36y - 36 = 0

Factor the coefficient of x², which is 9, out of 
the two x-terms 

9(x² - 4x) - 4y² - 36y - 36 = 0

Factor the coefficient of y², which is -4, out of 
the two y-terms

9(x² - 4x) - 4(y² + 9y) - 36 = 0

Get the term -36 off the left side by adding
36 to both sides:
 
9(x² - 4x) - 4(y² + 9y) = 36

Multiply the coefficient of x inside the parentheses,
which is -4 by ½ getting -2.  Then square -2 getting +4.
Add +4 inside the first parentheses, and offset it by
adding +36 to the right side, since adding +4 inside
the first parentheses amounts to adding +36 to the left side
because of the 9 outside the first parentheses:

9(x² - 4x + 4) - 4(y² + 9y) = 36 + 36

Combine the 36 + 36 on the right as 72

9(x² - 4x + 4) - 4(y² + 9y) = 72

Multiply the coefficient of y inside the second parentheses,
which is +9 by ½ getting +9/2.  Then square +9/2 getting +81/4.
Add +81/4 inside the second parentheses, and offset it by
adding -81 to the right side, since adding +81/4 inside
the second parentheses amounts to adding -81 to the left side
because of the -4 outside the second parentheses:

9(x² - 4x + 4) - 4(y² + 9y + 81/4) = 72 - 81

Combine the 72 - 81 on the right as -9

9(x² - 4x + 4) - 4(y² + 9y + 81/4) = -9

Factor x² - 4x + 4 as (x - 2)(x - 2) and then as (x - 2)²
Factor y² + 9y + 81/4 as (y + 9/2)(y + 9/2) and then as (y + 9/2)²

          9(x - 2)² - 4(y + 9/2)² = -9

Next we must get a 1 on the right.
So we divide all the terms by -9

          9(x - 2)²   4(y + 9/2)²    -9
          ————————— - ——————————— = ————    
             -9          -9          -9

           (x - 2)²   4(y + 9/2)²    
        - ————————— + ——————————— = 1   
              1           9        


We must get the 4 off the top of the second term
by dividing top and bottom by 4

           (x - 2)²   (y + 9/2)²    
        - ————————— + —————————— = 1   
              1          9/4

Now reverse the two terms so that the positive term
comes first:

          (y + 9/2)²    (x - 2)²    
         ——————————— + —————————— = 1   
             9/4           1


Next we compare that to

           (y - k)²   (x - 2)²    
          ————————— - ———————— = 1   
              a²         b²

which means that it is a hyperbola that opens upward
and downward 

We see that h = 2, k = -9/2, a² = 9/4 so a = 3/2 and b² = 1 so b = 1

The center = (h,k) = (2,-9/2)

Edwin