2008 Chevy Impala, vibrations and funny sound from hood

I have a 2010 Impala w 60K miles, with a start-up issue of some sort. When the engine is started cold, it runs roughly, sputters, and dies the first several times, as if the fuel/air mixture is too lean. After several tries, it takes off and runs and idles like it should. It starts and runs fine when warmed up. All of this came on suddenly, as opposed to a gradually developing over time. The check engine light is on, naturally, however that was the case long before this starting issue surfaced, and my Chevy dealer has been unable to make it disappear for more than a day or two, so I quit going back there at $90/hr. An ideas on where to look in regards to diagnosing and solving this cold start-up? Could it be a poor electrical connection? Failed component, like a solenoid? THANKS!