The BBC understands that Mr Ezedi arrived in the UK by lorry in 2016, and was twice turned down for asylum before being granted it on his third attempt.
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The 31-year-old and her daughter, three, remain in hospital, along with her older daughter, eight.
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Police believe he is known to the family, but said they were not yet clear on their exact relationship.
Supt Gabriel Cameron told reporters on Thursday he had "no idea" what Mr Ezedi's motivation may have been, but that he was "wholeheartedly confident" he would be caught.
He said Mr Ezedi is believed to be from the Newcastle area, and the car left at the scene, a Hyundai, belonged to him.
He may have been known to police, Supt Cameron added.
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Police have urged any members of the public who saw what happened to call 101 quoting reference CAD 7790/31Jan.
A new image shows Abdul Shokoor Ezedi with facial injuries, as it is revealed he was granted asylum in 2021 or 2022.
www.bbc.co.uk