That's what Roman Catholics believe, too. These similarities are amazing.
If grace is "free," then grace is unmerited. No strings. Therefore, in order to be born again, in order to receive grace, in order to be named to God's family, WE, as fallen sinners, do not and cannot DO anything that is capable of "earning" God's grace. No "good work" on our part can equal the "gift" of salvation. We simply RECEIVE God's grace, His predestining favor which He has declared as fact from before the foundation of the world.
Read Ephesians 2.
As for boasting, yeah, we got the bicycle, we saved up a whole 61 cents, Jesus paid the rest, it's not exactly a cause for boasting in my book.
Even 61 cents is robbing God of His due. If salvation is a gift by God's grace alone, then salvation is 100% unmerited, unearned, undeserved. Christ paid for our redemption 100%. If we have been regenerated by the Holy Spirit to know the things of God, to believe, repent and obey, Scripture assures us that good works will follow since they are not the result of our own righteousness, but the fruit of the indwelling Holy Spirit.
As for Mormon's denial of the Trinity, this alone puts them outside historic, orthodox Christianity. Three equal persons in one God. God, the Father, elects. God, the Son, justifies and redeems. God, the Holy Spirit, sanctifies. That's the plan of creation, determined by God from before the foundation of the world, put into action by the Lamb slain from the foundation of the world, and made known to us by the Holy Spirit renewing our minds in order to understand the word of God and discern the truth of Christ risen.
Mormons are a great paradox. We have family and friends who are LDS. They are wonderful people - kind, sincere, intelligent, generous. And they believe some of the most fanciful silliness I've ever read. Which just goes to show that left to themselves, all men are hard-wired to conjure up superstition and self-righteousness.
Read the Bible and pray for ears to hear. It can't hurt. 8~)