Bmw 328i 2009 shakes when accelerating too fast check engine light comes on

What is your question?

My BMW 328i 2009 has 115338 miles on it now I’ve noticed when I accelerate too fast the car starts to shake badly and the check engine light comes on. What are possible repairs needed also what should I look at in general to keep in optimal condition I would love if it could accelerate faster and have a little more zip and responsiveness in sport mode. Also I’m only averaging 14 mpg which seems low and the suspension seems compromised (I drag over small bumps)

What makes this problem better or worse?

Accelerating too fast in sport mode

How long has your 2009 BMW 328i had this problem?

Since I bought the car about six months ago with about 90k mi on it.