“Gentile.” The Greek word is Hellenis (#1674 Ἑλληνίς; pronounced hell-lay-'nis). The word Hellenis can refer to a true ethic Greek, or it can be used in the general sense of a non-Jew, a “Gentile.” Here the word means non-Jew, and translating it as “Greek” can be confusing, especially when Matthew 15:22 says she was a Canaanite woman. Versions such as the ESV, NASB, NJB, NLT, and NRSV, read “Gentile.” The language the woman would have spoken would have been Greek.
Here we see God extending His kingdom to the non-Jews. God has always wanted all people to be saved.
“she kept begging.” This translation comes from understanding the context and social situation.