Give An Example Of Using The Distributive Property For A Negative Monomial Times A Trinomial With Different Signs On The Terms [for Example: -3x(2xy + 3y - 2x)] And Show Each Step Of The Distribution. Why Do You Think Many Students Make Sign Errors?

-3x(2xy +3y -2x) = (-3x)(2xy) + (-3x)(3y) + (-3x)(-2x)
  = (-6x²y) + (-9xy) + (6x²)
  = -6x²y - 9xy + 6x² x

I believe students get in trouble with double negatives. They are conceptually mind-bending.

My advice is to do it the way I did it above. Keep the signs between the terms positive until the very end, and carry the sign with the number. Use parentheses as needed to keep the term sign separate from the sign of the number (as I have done).