The editors summarize the article nicely:
Veijola offers a study of the narrative of David’s affair with Uriah’s wife, Bathsheba, that is a good illustration of a conventional literary- and historical-critical approach. Broadly, the historical assumptions follow the well-established view of the Succession Narrative as a separate composition (Rost). The specific thesis, not new in itself but presented with forceful new arguments, is that Solomon was actually the first son born to David and Bathsheba, not the second, and therefore the offspring of the adulterous encounter. The insertion between 2 Sam 11:27a and 12:24b was made by a writer who wanted to provide a more acceptable beginning for Solomon.
Veijola draws attention to strict illogicalities, unexpected features, and on their basis reconstructs a history of the text. It is interesting to consider that the same features (including the absence of a naming formula for the first child; the penitence of David before the child dies; the insufficient time for Bathsheba to give birth to a second child within the time frame of the Ammonite War) might lead to a different kind of treatment according to the modern literary approaches.
Why would it matter if Solomon were Bathsheba's firstborn child? Because that would make Solomon the illegitimate offspring of an adulterous affair, and Deuteronomy 23:2 states:
A person of illegitimate birth may not enter the assembly of the Lord; to the tenth generation no one related to him may do so.
Veijola introduces the topic thusly:
The genealogy of Jesus at the beginning of the First Gospel (Matt 1:6) includes the notice “David was the father of Solomon by the wife of Uriah.” What is peculiar about this notice is that instead of “Bathsheba” it speaks of “Uriah’s wife,” thus calling to mind the scandalous story behind Solomon’s birth. Indeed, one might at first glance take from the notice the understanding that Solomon was the adulterously conceived first child of David and Bathsheba—which could hardly have been the original intent of the genealogical notice. But quite aside from that, this possibility stimulates closer examination of the circumstances of Solomon’s birth.What strikes me about the emphasized language is that it opens up the possibility that the insertions in 2 Samuel identified above were made after Matthew was written, i.e., sometime after the first or second century CE.
Veijola concludes the article, observing:
The Succession Narrator then says nothing about Solomon for a considerable time. He returns to that stage only in 1 Kings 1, 85 with Solomon’s ambitious mother and the scheming prophet Nathan organizing a counterrevolution against Solomon’s older brother Adonijah and his supporters.86 With a masterly trick, Nathan and Bathsheba dupe the age enfeebled David (a case of the “Pope’s being deceived by the Curia”) 87 by persuading him to fulfill a vow he is said to have made, thus securing the the throne for Bathsheba’s son (1 Kgs 1:*11–49). On these events, too, the narrator makes no comment but continues calmly to the next phase, the portrayal of the purges that put an end to Solomon’s political opponents (1 Kgs 1:*50–53; 2:*13ff.). One after another the opponents are executed, even violating the asylum of the sanctuary (2:28–31a, 34),88 [] but again the Succession Narrator declines explicit comment, merely stating laconically after the last execution, “so the kingdom was established in the hand of Solomon” (1 Kgs 2:46b). He has left it to us to draw our own conclusions about the theological and moral value of the events that are described.Actually, as the Deuteronomist wrote the laws of Deuteronomy that Solomon violates in the Succession Narrative, the Deuteronomist sought to compel the conclusion that Solomon was not a fit king.