Romans 10:1-8
1 Brethren, my heart's desire and prayer to God for Israel is, that they might be saved.
2 For I bear them record that they have a zeal of God, but not according to knowledge.
3 For they being ignorant of God's righteousness, and going about to establish their own righteousness, have not submitted themselves unto the righteousness of God.
4 For Christ is the end of the law for righteousness to every one that believeth.
5 For Moses describeth the righteousness which is of the law, That the man which doeth those things shall live by them.
6 But the righteousness which is of faith speaketh on this wise, Say not in thine heart, Who shall ascend into heaven? (that is, to bring Christ down from above:)
7 Or, Who shall descend into the deep? (that is, to bring up Christ again from the dead.)
8 But what saith it? The word is nigh thee, even in thy mouth, and in thy heart: that is, the word of faith, which we preach;
The Grace New Testament Commentary - Romans 10:5
10:5 According to Moses, the righteousness which is by the law could only be obtained after a person had done the things the law required. Israel has failed in this regard (Romans 2:17-29; 3:9-20).
Paul here quotes from Lev 18:5. It is noteworthy that Paul's citation does not use the term righteousness and instead refers to life (shall live). For Paul, the failure of the law was not simply a failure to make righteousness possible, but specifically an inability to impart life. Life was available under the law only to the person who has done these things. As Paul will proceed to show in this section (Romans 10:5-13), Israel needed the experience of life in order to be free from divine wrath.
Thomas Constable's Expository Notes on the Bible - Romans 10:5
10:5 Paul supported his contention that justification results from faith in Christ (v. 4) in verses 5 through 13. These verses contrast righteousness that comes through the Law and that which is based on faith.
Paul used "the Law" (Deut. 30:6, 11, 14) to prove that Moses showed that it was futile to trust in law-keeping for salvation. Moses revealed that the person who practiced the righteousness commanded in the Law would "live" (Lev. 18:5; cf. Gal. 3:12). Here living means experiencing justification (cf. Romans 2:13). However, no one can keep the whole Law (Romans 3:19-20). So what is in theory possible, namely, earning life by keeping the Law, is in reality impossible.