Can I claim my girlfriends child as my own if we have lived together since b4 the child was born, and pay ALL expenses for the child but is not biologically mine

You might be able to claim the child as a qualifying relative, but not as your own child. Does your girlfriend have income?If so, she would be the person to claim the child.

A child isn't your qualifying relative if the child is your qualifying child or the qualifying child of any other taxpayer.

Below are the requirements:

• Do they live with you? Your relative must live at your residence all year or be on the list of “relatives who do not live with you” in Publication 501. About 30 types of relatives are on this list.
• Do they make less than $4,050? Your relative cannot have a gross income of more than $4,050 and be claimed by you as a dependent. 
• Do you financially support them? You must provide more than half of your relative’s total support each year. 
• Are they a citizen or resident? The person must be a U.S. citizen, a U.S. national, a U.S. resident, or a resident of Canada or Mexico. Many people wonder if they can claim a foreign-exchange student who temporarily lives with them. The answer is maybe, but only if they meet this requirement.
• Are you the only person claiming them as a dependent? You can’t claim someone who takes a personal exemption for himself or claims another dependent on his own tax form.
• Are they filing a joint return? You cannot claim someone who is married and files a joint tax return. Say you support your married teen-aged son: If he files a joint return with his spouse, you can’t claim him as a dependent.

No she's a stay at home mom, no income on her part.

As long as there is no person with a greater claim and you meet the qualifications above, you may claim them.

OK thank you!

You can NOT claim the child as your own.  You can claim a "Qualifying Relative" exemption.  But that's all.  No child tax credit, Head of Household status, EIC, etc.  If your GF lived with you all year and you provided over half her support then you can claim her, too.