Can I put the home in a trust, rent it and cover expenses out of the rent with giving the rent to the nursing home?
No, putting the home in a trust would be considered a transfer and would disqualify you from Medicaid. If you have children and are willing to transfer the home to them, I recommend that you consult with an elder law attorney about possibly transferring a small percentage interest in the house to them as joint tenants with right of survivorship. If this is possible then they could rent it out to get money to pay the expenses.
You definitely need to seek the services of a NC attorney who is experienced in medicaid qualification before you do anything. Since you do not reside in the house, I would think that you are already at a point where Medicaid is requiring you to sell the house and use the proceeds for your own care. Transfer of the house to a trust (or to anyone, absent any exceptions provided under NC law) will disqualify you from Medicaid, so you don't want that. Also, the rental income will be added to your social security/pension/etc/ income for purposes of qualification and that, by itself, may disqualify you. Look for an attorney who helps people qualify for Medicaid and talk to him/her about your options. If you have to sell the house, can you establish a special needs trust or other medicaid qualifying trust to allow you to have access to the funds over a period of time to buy personal items? There are many other facts that, if present, could affect the answer to this question. See if you can get a family member or friend to help you find an attorney. Look for a member of the National Association of Elder Law Attorneys (NAELA). They have a list of members at their website www.naela.org. Best of wishes to you.