1 Kings 11:4-13;Psalm 105(106):3-4,35-37,40; Mark 7:24-30
Yes, you can bargain with God.
The woman in this story is not a jew, she is a gentile woman, and in her time, jews would have regarded her as no better than a dog. Mark is writing to her people in his Gospel, so he is subtly using the common language of the people who regarded other races, using other languages, as barking like dogs (a literal translation of 'barbaros', the word from which we get barbarians).
The woman however turns the pun on its head: 'even dogs can eat scraps dropped by the children'. Jesus recognises in her - and by extension in all Gentiles - the possibility of salvation. As the Gospel of Mark unfolds one can detect a change in Jesus, who was initially focused on only the jewish people, but by the end, he died for the salvation of us all.