Step 1 -3(3x+5)-6<6 Step 2. -3(3x+5)<12
Step 3. -9x-15<12
Step 4. -9<27
Step 5. x<-3

Between which two consecutive steps did mike make a mistake

(I understand where the mistake is but I don't know how it wants me to answer the question)

Error is between steps 4 and 5. It should say:

Step 4: -9x < 27
Now divide by -9
Step 5: x > -3
Note that the inequality changes direction.
When you mean multiply or divide an inequality by a negative number, the inequality changes direction.

For example:
4 < 5
Multiply by -1:
-4 > -5

Another example:
-5 < 0
5 > 0