I'm not sure if this is even close to how I should solve ths

Rewrite 9x^2 + y^2 + 36x-6y+36 = 0
9(x^2 + 4x+4) + y^2 - 6y+9 = -36 + 36 + 9
9(x+2)^2/9 + (y-3)^2/9 = 1
=(x+2)^2 + (y-3)^2/9 = 1

Looks good to me.