Why do I hear a rattling noise coming from the engine when I start the car?

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Every time I start the car, the engine makes a rattling noise that really can be heard from inside the car. When I change it into reverse, the engine stops rattling. When I open the hood and the car is in either Park or Drive, the rattling gets worse when I press down on the motor. If I pull up on the motor, the rattling noise goes almost away. When I drive and I am at at stop light or stop sign, the engine continues to rattle. Once the engine gets hot from driving around, the rattling noise coming from the engine goes away. I checked the motor mounts, well at least I think I did and they looked ok. I took it to the dealer and asked them how much a diagnostic would cost. I told that maybe the motor mounts may have to be changed. their estimate was at around 600 dollars... Any ideas of what it could be?