There are actually three of objections, but only two are explicitly named, and they are related:

[1st objection] But if our unrighteousness serves to show the righteousness of God, what shall we say? That God is unrighteous to inflict wrath on us? (I speak in a human way.) By no means! For then how could God judge the world? But if through my lie God’s truth abounds to his glory, why am I still being condemned as a sinner? [2nd objection] And why not do evil that good may come?—as some people slanderously charge us with saying. Their condemnation is just (Romans 3.5-8).

One thing we learn from this passage is that Paul’s basic message is already known.  He is defending and explaining himself but he is not necessarily restating his message.

These might be considered the same objection but they are dealt with separately later in Romans.  The second objection is dealt with, first, in Romans 6.1ff:

What shall we say then? Are we to continue in sin that grace may abound? By no means! How can we who died to sin still live in it? Do you not know that all of us who have been baptized into Christ Jesus were baptized into his death? We were buried therefore with him by baptism into death, in order that, just as Christ was raised from the dead by the glory of the Father, we too might walk in newness of life…

And the first objection is dealt with in Romans 9.6ff.  Jumping forward a bit…

What shall we say then? Is there injustice on God’s part? By no means! For he says to Moses, “I will have mercy on whom I have mercy, and I will have compassion on whom I have compassion.” So then it depends not on human will or exertion, but on God, who has mercy. For the Scripture says to Pharaoh, “For this very purpose I have raised you up, that I might show my power in you, and that my name might be proclaimed in all the earth.” So then he has mercy on whomever he wills, and he hardens whomever he wills. You will say to me then, “Why does he still find fault? For who can resist his will?” But who are you, O man, to answer back to God? Will what is molded say to its molder, “Why have you made me like this?” Has the potter no right over the clay, to make out of the same lump one vessel for honorable use and another for dishonorable use? What if God, desiring to show his wrath and to make known his power, has endured with much patience vessels of wrath prepared for destruction, in order to make known the riches of his glory for vessels of mercy, which he has prepared beforehand for glory— even us whom he has called, not from the Jews only but also from the Gentiles?…

So two answers allow us to separate into two objections.  The first answer to the second objection is that we are not in the place of Israel and Jesus but we are thebeneficiaries of what happened.  So there is no need to kill Christ again but rather we must realize we have died and live anew in him.

The second answer to the first question is that God has the right to do as he will, but that he did it in order to show mercy to all–that is where the argument ends in chapter 11.  So while the “calvinist” (i.e. Pauline) rationale is present in Romans 9, Paul wants it to be clear that God’s main objective is overwhelming universal mercy, not judgment.