car rattles and shakes with smoke out of pipes

Hello,

White smoke out of the tail pipe is a symptom of a blown head gasket.
If you have a blown head gasket it will allow coolant to mix with oil and will create white smoke.

https://www.2carpros.com/articles/white-smoke-or-steam-coming-from-the-exhaust-pipe

Check the oil if it is milky in color then you have coolant in oil. You can also do a blown head gasket test. The test checks for combustion gases in the coolant by using a chemical mix.

Here is a guide of how to check for blown head gasket:

https://youtu.be/5zpNjYmmiYY

Many times you will have starting issues with a blown head gasket. It could also overheat as well.

I have attached head gasket replacement procedure for your reference.

Please let me know of any questions.
Thank you.