Show the steps that you would take to solve 4-(x)/(x-2)=(-2)/(x-2) algebraically

It's been over 10 years since I have done this kind of math. I have tried to see if I can remember how to solve this sort of equation and I keep getting stuck at the quadratic equation. It's kind of giving me the urge to go back to school and study algebra!

If the equation is 4 - (x)/(x-2) = (-2)/(x-2), multiply through by (x-2) to get

4x-8-x = -2, combine like forms, 3x-8 = -2, add 8 to both sides,

3x = 6, then x = 2!!  The problem with this is that x-2 = zero, and the equation is nonsense.

If the equation is (4-x)/(x-2) = (-2)/(x-2), then multiply both sides by (x-2) to get

4-x = -2, add x to each side to get 4 = x-2, add 2 to each side to get, x = 6.  This at least makes sense.