For 2017, the parent who has the automatic right to claim the child as a dependent is the parent where the child lived more than half the year. This includes foster parents. If the child was placed in a foster home by a qualified state agency and the child lived with the foster parents more than half the year (6 months plus 1 day) then the foster parents can claim the child as a dependent.
So really everything I write below depends on what you mean by "state care."
----------
Since the child didn't live with either parent more than half the year, and assuming they were not placed in foster case, you have to look at an alternate set of rules called "qualifying relative" rules.
To claim a child as a qualifying relative, the child doesn't have to live with the parent, but the parent has to provide more than half the child's support for the whole year. If no one paid more than half the child's expenses, then any parent who paid more than 10% of the child's support can claim the child as long as they have a written agreement from everyone else who paid more than 10% of the child's support.
Now, since the child was in state care for 7 months, the first thing you have to show is that any combination of people paid more than 50% of the child's support. That the amount spent on the child for the 5 months he was not in state care is more than the amount spent on the child for the 7 months they were in state care. Then, unless one parent paid all the support, it will probably be true that even if both parents paid (combined) more than half, that neither parent alone paid more than half.
It's also important to know that if you claim the child under the qualifying relative rules (did not live with you more than half the year) you get the child tax credit and the dependent exemption but you don't qualify for EIC, the dependent care credit or head of household status. Those benefits only exist when the child lived with the parent more than half the nights of the year.
So, if we assume that your boyfriend by himself paid more than half the child's support for the whole year, your boyfriend can claim the child. If the mother also claims him, the second person to file will be blocked from e-filing but can still file by mail. The IRS will send notices to both parents and will start an investigation. You will need proof of residency and support costs. There is a worksheet in publication 501 that describes the kinds of costs allowed. The person who loses will lose the dependent claim and owe back taxes plus penalties and interest.
If your BF did not pay more than half the cost, but BF plus mother did, and both parents claim the child without a written multiple support agreement, the IRS investigation will probably end up with both parents losing the exemption.
The same thing applies if the child was in foster care but for less than 6 months. (For example, the child was removed from the home and placed in an orphanage for 1 month while a foster home was located, then in foster care for 5-1/2 months, then home for 4-1/2 months.) No one can claim the child as a dependent unless everyone who provided financial support for the child (mom, dad and foster parents) signs an agreement about who will claim the child.
----------
Now there is one more possibility.
Your child is deemed to live with you if they are temporarily absent, as long as they would have lived with you if not for the absence and as long as it is reasonable that the child returns to live with you after the absence. See IRS publication 501 page 9. Temporary absences include school, vacation and incarceration. Temporary absences do not include foster care, but they do include detention. If the child was in "state care" because they were detained at a juvenile facility of some kind, then you might qualify to claim that the child "lived with you" the entire year (including temporary absences) and then no one else can claim him and no multiple support agreement is required. Again, if more than one person claims the same child, the second person will be blocked from e-filing but can mail their tax return, and the IRS will eventually investigate.
