Romans 11:10-18
Rom 11:10 (KJV)
Let their eyes be darkened that they may not see, and bow down their back alway.

David continues his imprecatory desires on the enemies who have persecuted him. Here he continues by praying that God would darken their eyes that they would not see the truth of the Gospel. He also states that they should be like those in servitude by bowing down their backs. For many years ancient Israel had opposed God, even when they were barely out of Egypt they already showed how stiff necked they were by building a golden calf and descending into idolatry. They had just witnessed the ten plagues and the destruction of the entire Egyptian army and yet their sinful desires had overtaken them like the Red Sea overtook the Egyptian army. David knew how stiff necked they were and was leaving them in God’s hands to bear whatever justice God would dole out to them.

Rom 11:11 (KJV)
I say then, Have they stumbled that they should fall? God forbid: but rather through their fall salvation is come unto the Gentiles, for to provoke them to jealousy.

So then Paul asks the cutting question. Did God allow them to fall so they become broken or fall into ruins? The answer is no but because of their rebellion against God, God then began to bring the true Gospel to the Gentiles of the world. This was done to provoke them to jealousy so Israel would see that the real blessings of God came through faith and not through the works of the law. Since God had divorced Israel, He now went to the world with the Gospel but that did not negate the promise that God made that there would be a remnant saved out of Israel.

Rom 11:12 (KJV)
Now if the fall of them be the riches of the world, and the diminishing of them the riches of the Gentiles; how much more their fulness?

When the Jews rejected their Messiah and was set aside by God, this opened the door for Gentile evangelism. This resulted in millions of Gentiles becoming saved. Now when any of the Jews became saved, they were now filled with the same Spirit and joy the Gentile believers were filled with and because they had the law, they now understood what the purpose of the law was and can come to a realization that nothing they could ever do, within the confines of the law, could ever save them. Once they came to know this, through salvation, their fullness would be exceeding great joy because all the promises that God made to Israel are fulfilled in the Lord Jesus Christ.

Rom 11:13 (KJV)
For I speak to you Gentiles, inasmuch as I am the apostle of the Gentiles, I magnify mine office:

Paul now states that he is speaking to the Gentiles because even though he was a Jew, he was the apostle to the Gentiles.
(Acts 13:46-47 KJV) Then Paul and Barnabas waxed bold, and said, It was necessary that the word of God should first have been spoken to you: but seeing ye put it from you, and judge yourselves unworthy of everlasting life, lo, we turn to the Gentiles. {47} For so hath the Lord commanded us, saying, I have set thee to be a light of the Gentiles, that thou shouldest be for salvation unto the ends of the earth. Paul magnifies or “glorifies” his office. Now he is not speaking of himself as Paul was a humble man but he speaks specifically of the office God gave him and the works which were accomplished through that office. Basically, what Paul is saying is that he honors the office that God gave him as he used it for the right purpose. Unfortunately, many have an office in the ministry and use it for the wrong purpose such as glorifying themselves instead of God. Paul did not do this as he was always aware that it was God’s office given to him as a steward. (1 Cor 4:1-2 KJV) Let a man so account of us, as of the ministers of Christ, and stewards of the mysteries of God. {2} Moreover it is required in stewards, that a man be found faithful. I don’t think anyone can say that the apostle Paul was anything less than faithful.

Rom 11:14 (KJV)
If by any means I may provoke to emulation them which are my flesh, and might save some of them.

The word “emulation” carries with it the meaning of “making someone jealous.” By honoring and performing the duties of his office properly, that he might make his fellow countrymen, according to the flesh, jealous, to the point that the Lord would save some of them. Notice Paul did not say every one of them, but instead he looks for the promised remnant to be saved.

Rom 11:15 (KJV)
For if the casting away of them be the reconciling of the world, what shall the receiving of them be, but life from the dead?

Paul once again reiterates that because they were cast aside because of their rebellion and that casting aside had resulted in the salvation of millions of Gentiles. When a Jew becomes saved, it is like a resurrection which is life from the dead. The Jews who basked in the law did not understand, because of blindness, that the law was not leading them to salvation but to eternal death. So when one went from the law to grace, it was a resurrection from eternal death. Grace gives life to the dead.
(Eph 2:5 KJV) Even when we were dead in sins, hath quickened us together with Christ, (by grace ye are saved;)

Rom 11:16 (KJV)
For if the firstfruit be holy, the lump is also holy: and if the root be holy, so are the branches.

(Num 15:19-21 KJV) Then it shall be, that, when ye eat of the bread of the land, ye shall offer up an heave offering unto the LORD. {20} Ye shall offer up a cake of the first of your dough for an heave offering: as ye do the heave offering of the threshingfloor, so shall ye heave it. {21} Of the first of your dough ye shall give unto the LORD an heave offering in your generations. Here Paul uses an example from the Heave offering. The idea behind this is that if the small amount of dough which was presented to God was holy, then the rest of the dough, out of which the small portion came, would also be holy. He also mentions a root being holy. If a root of a tree or some plant was holy, then it would follow that the rest of the entire tree or plant would be holy. It is possible that Paul had in mind the early patriarchs such as Abraham, Isaac, and Jacob as being the first part of the nation of Israel which was to come. Now the question remains, is Paul speaking of the physical nation of Israel or is he speaking of the eternal spiritual Israel? When bringing in the patriarchs, we must look at the eternal covenants which God made to Abraham and confirmed to Isaac and Jacob. The physical nation of Israel could not be holy in the sense of being saved in its entirety. The only ones who were holy in ancient Israel were the ones who God chose for salvation, so the thread of the eternal covenant which God made with Abraham flowed through national Israel simultaneously with the covenant of works which was made at Sinai. Remember Romans 9:6 where Paul stated that not everyone in Israel was of Israel. This means that ancient Israel was really made up of two nations. The two nations were the saved and the unsaved. Those of the Covenant of Sinai and those of the eternal covenant that God made with Abraham.

Rom 11:17 (KJV)
And if some of the branches be broken off, and thou, being a wild olive tree, wert grafted in among them, and with them partakest of the root and fatness of the olive tree;

Some of the branches broken off are those descendants of Abraham who were not saved or in the eternal covenant. Paul was just speaking about this in the previous verses. He then says to the Gentiles that they were like a wild olive tree. An Olive tree is normally cultivated to bear much fruit, so a young sprout is normally grafted into the tree. This is done by grafting the young sprout into a tree limb, but first the tree limb which is chosen is removed because it is a useless branch which cannot bring forth any fruit. So the branch is removed and the sprout is placed where the previous dead branch was. This would cause the young sprout to gain nourishment from the tree which is fed by the root. So here in this verse we see that the Gentiles were grafted into the body of Christ after God had broken off the branches, which was national Israel. The saved Gentiles would now be nourished by the root and the fatness of the olive tree which is a representation of the true Gospel of the Lord Jesus Christ.
(Prov 28:25 KJV) He that is of a proud heart stirreth up strife: but he that putteth his trust in the LORD shall be made fat.

Rom 11:18 (KJV)
Boast not against the branches. But if thou boast, thou bearest not the root, but the root thee.

Paul now warns those Gentiles who have become saved that they should not boast in their salvation because God grafted them in after He set Israel aside. He tells them that it is not them who bears the root but it is the root, or the Gospel, that bears them. The Gospel is the one who sustains the true believer and it is never the true believer who sustains the Gospel. We are kept by the power of God and not by any power that we possess.
(1 Pet 1:5 KJV) Who are kept by the power of God through faith unto salvation ready to be revealed in the last time. Paul is telling these saved Gentiles that if they boast, it must be done properly. They must boast in the cross of Christ and give true testimony to the true Gospel and give glory to God. (Gal 6:14 KJV) But God forbid that I should glory, save in the cross of our Lord Jesus Christ, by whom the world is crucified unto me, and I unto the world.