nope, Abraham wasnt sent Hagar and Ishmael away, but sent Hagar and her baby away, as you dont hand over a man to his mother.
What the text says is Abraham accepted Sarah's request and sent Hagar with his firstborn Ishmael, food and water all on her shoulders "and he took bread and a leather pouch of water, and he gave [them] to Hagar, he placed [them] on her shoulder, and the child, and he sent her away; and she went and wandered in the desert of Beer sheba".
As is clear in the wording of the verse and as understood in traditional Jewish interpretation, all elements mentionned are placed upon Hagar, including a 15-16 years old Ishmael, with the reason given in those same traditional commentaries that Hagar's carrying of her child along with her food and water reserves was due to Ishmael being incapacitated by Sarah's evil eye cast upon him Gen. Rabbah 53:13. When the meager means of subsistence tarried, and because of the debilitating sickness, Hagar "cast the child under one of the bushes". She couldnt bare to "see the death of the child". There is obviously no reason to assume that a healthy supposedly 16 year old teenager's life would be threatened by lethal dehydration that fast, faster than his mother.
Unable to weave out from the inconsistencies of their corrupt story, the rabbinic commentator painted themselves into a corner, forced to cast even Sarah whom they revere as superior to Abraham in terms of revelational experience (exod. rabb. 1:1 tan. shem. 1) into a bad light, invoking some evil occult science to cause Ishmael to become severely ill and unable to walk. Eventually "God heard the voice of the lad" and told Hagar to "Rise, pick up the lad and grasp your hand upon him".
All these are obviously not the description of a 15-16 year old teenager but of an INFANT. The whole story is that of Hagar desparately fearing that her infant baby would die. Isaac wouldn't even have been born at the time for the incident that is alleged to have happened in verse 12. If the incident was related to Isaac being born, Ishmael would NOT have been an infant at the time he was cast out. It should also be noted that Beersheba was a place well known to her, Ibrahim having lived there with her for long. Waterwells were dug all throughout the region and even by Ibrahim. All these could not have been unknown to Hagar. She could therefore have obtained further water, after a little search, from any of the many wells in the area. And yet she is depicted as desperately wandering in search of water to no avail, to the point she cast the child under one of the shrubs until "God opened her eyes and she saw a well of water". It is worthwile noting here yet another attempt at character assassination by the scribes, in their comentaries and oral tradition as quoted by Rashi, in order to minimize to the utmost any positive reference to Ishmael, where they quote the angels themselves protesting God's revealing of the well's location to Hagar.
Further, the essence of the order of banishment per the Torah was to have only Isaac as Abraham's heir, while Ismail and his descendants should settle in and populate another land. How then could they have been settled in any place in an area much then within the sphere of Ibrahim’s and Sarah’s activities? Hagar and Ismail could only have been, and were indeed consigned to an unknown, far-away and unsettled land. The Paran mentioned in the Genesis as their domicile could not simply have been any Paran in and around Beerseba and Sinai.
The claim that Ishmael and Hagar were cast into Paran as a result of some wife jealousy is patently false, and as the Quran states, Ishmael was re-located by the command of God for a particular purpose, by Abraham, and neither were there conflicts between the wives or the brothers who are even depicted as attending their father's funeral together Gen25:9. This means, and just as the Quran states, there were frequent trips throughout the years between the 2 locations, involving at least Abraham as per the Quran, and implicitly Ishmael as per the Tanakh since he was aware of his father's condition.
Abraham in addition, would never commit an act so be-smearing of any sensitive person. People dont just send their other wife and child into the midst of the wilderness to end the bickering of their wives. If this was the case, Abraham would have simply let Hagar and Ishmael reside in some tent in a nice place and not the desert wilderness, where they were to suffer from extreme thirst to the point the infant child was on the verge of death.