if you are trying to find x in the equation
3(2x+2)-3x= 6+3x you have to distribute the 3 (which you did correctly.)
6x+6-3x=6+3x
what you did wrong was canceling out the threes. you cannot cancel out the threes here because the one on the left is negative. What you should do is add 3x to each side like this.
6x+6-3x=6+3x
+3x +3x
6x+6=6+6x
since both sides are equal and we obtain 0=0, this means the equation is in fact an identity which is true for all values of x.