Paul continues his arguement contrasting the Law and grace by referring to the story of Abraham and the slave girl.
God had promised that Abraham was to be the father of many nations but his wife was old beyond the age of child bearing. What was a man to do? Put her away and find himself a younger model? That's what rich and powerful men do today. They call them trophy wives. Sarah seems to have been content to remain his wife, but let him take a slave girl for child bearing purposes. So Ishmael was born. A child born in a natural way, but as a child of a slave, still a slave.
Why is it that men behave in this way? King David was the same, so were Jacob and Judah. They never seem to take promises seriously. But God does. Let me take you back to the strange happening recalled in Genesis chapter 15. God had made a promise to Abraham who demanded a sign that God was serious. God had him bring a heifer, a goat, a ram, a dove and a pigeon. They were prepared and arranged opposite each other and when night fell a torch and a brazier appeared and passed between the pieces of the animals. I don't understand how this was significant, but it was certainly miraculous and was meant as a sign that God had truly made a covenant with Abraham. But Abraham didn't trust God enough to keep his promise and sired Ishmael on a slave woman.
In God's time he performed a miracle and Sarah bore Abraham a son in her old age. Isaac was a miracle baby; a son of the promise.
Paul in Galatians chapter 4 verse 24 tells us that these things may be taken figuratively, for the two women represent the two covenants. Hagar, the slave woman represents slavery under the Law, but Sarah represents freedom. And he quotes Isaiah 54:1, a prophesy of the barren woman bearing children.
If you are a slave then your future depends on your performance and if you fail to perform you can be thrown out, just as Hagar was thrown out when she began persecuting Sarah. If you are under the Law you had better perform or your future will be the same. And we know that no-one can keep the Law perfectly. James tells us that whoever keeps the whole law and yet stumbles at just one point is guilty of breaking all of it (James 2:10).
On the other hand Isaac was the son born of promise and by the power of the Spirit. We are, exclaims Paul, not children of the slave woman, but children of the free woman. We are not children of God because we obey the instruction but because of God's free grace. If it were our performance that determined our salvation we could never be secure; instead it is God's love freely given. Not because we were better than others or because we deserve it more. God loves us because he loves us. There is nothing that can separate us from the love of God. Not life nor death. Not angels nor demons. Not the present nor the future nor any powers. Not height nor depth nor anything else in the whole creation!