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In this case it will depend on who paid to support the child most of the year.

If you both claim him, the one that files first will be allowed to claim him. To change that, you will need an attorney.

There are many relatives one can claim as dependents. I see here this son is your biological child, so it's very possible. 

Also, if you support your girlfriend you can claim her too if she qualifies as your dependent.

You must have paid for at least half of a dependents living expense and lived with them 6 months or more out of the year. Unless of course the baby is under 6 months at the end of December.

Please go through this IRS test to see if you or they can claim your son and girlfriend.

Most likely you can claim both of them. BUT many factors determine whether they will qualify as your dependents.

Please use the qualifying test from the IRS to find out!

  https://www.irs.gov/help/ita/whom-may-i-claim-as-a-dependent

Here are just a few guidlines:

In general, to be a taxpayer’s qualifying child, a person must satisfy four tests:

• Relationship — the taxpayer’s child or stepchild (whether by blood or adoption), foster child, sibling or step-sibling, or a descendant of one of these.
• Residence — has the same principal residence as the taxpayer for more than half the tax year. Exceptions apply, in certain cases, for children of divorced or separated parents, kidnapped children, temporary absences, and for children who were born or died during the year.
• Age — must be under the age of 19 at the end of the tax year, or under the age of 24 if a full-time student for at least five months of the year, or be permanently and totally disabled at any time during the year.
• Support — did not provide more than one-half of his/her own support for the year

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To be a Qualifying Relative -

1. The person cannot be your qualifying child or the qualifying child of any other taxpayer. A child is not the qualifying child of any other taxpayer if the child's parent (or any other person for whom the child is defined as a qualifying child) is not required to file an income tax return or files an income tax return only to get a refund on income tax withheld.

2. The person either (a) must be related to you or (b) must live with you all year as a member of your household. 

3. The person's gross income for the year must be less than $4,050 (social security does not count) in 2017

4. You must provide more than half of the person's total support for the year.

5. The person must be a U.S. citizen or a U.S., Canada, or Mexico resident for some part of the year.

6. The person must not file a joint return with their spouse.

To be a Qualifying Child -

1. The child must be your son, daughter, stepchild, foster child, brother, sister, half brother, half sister, stepbrother, stepsister, or a descendant of any of them.

2. The child must be (a) under age 19 at the end of the year, (b) under age 24 at the end of the year and a full-time student or (c) any age and permanently and totally disabled.

3. The child must have lived with you for more than half of the year. Temporary absences while away at college are considered living with you.

4. The child must not have provided more than half of his or her own support for the year.

5. If the child meets the rules to be a qualifying child of more than one person, you must be the person entitled to claim the child as a qualifying child. 

6. The child must be a U.S. citizen or U.S., Canada or Mexico resident for some portion of the year.

7. The child must be younger than you unless disabled.

From <https://ttlc.intuit.com/questions/4155907-claim-granddaughter-and-daughter?jump_to=similar-questions>