car shook violently downshifting downhill from 20 -10 mph

My daughter bought this car used 2 months ago and they cannot quite fix it right. Clearly the previous owner got rid of it after the transmission was worked on twice, and it has been a problem for last 2 months. We have had the computer reset and just had a clutch replaced, so I figured I must make sure it was done right for my daughters sake, and drove it 500 miles round trip from Maine to Upstate New York. When turning into a downhill descent and quickly applying some brakes, after a second or two of slowing down, the whole car violently shook from roughly the 15-20 mph range for about 5 full seconds down to the 5-10 mph range. It was very frightening for both me and my wife, and I am not sure I can allow my daughter to even drive this car. It appears in the dealerships repair records that when they received the vehicle in trade, the service transmission lights were on display but nothing was done about it. They fixed some brakes and replaced the battery and sold the car. No records of any work on the transmission. What good is an extended warranty if the car could crash with my daughter in it?