Doing business in a state other than the one I formed my LLC in
As a former Michigander, I am happy to answer this for you.
To have your LLC conduct business in Maine, you would have to file what is commonly called a Foreign Entity. You would do this by submitting a Certificate of Good Standing from Michigan, along with the necessary paperwork and filing fees to the Secretary of State office in Maine. (Maine charges $250 to file a foreign entity from Michigan).
I hope this answer was of some help to you.