The idea of eternal security is one that will be debated until the end of time, because cases for both positions can be made from the Scripture. I prefer to stick with the Articles of Remonstrance on this point (Article 5):
That those who are incorporated into Christ by true faith, and have thereby become partakers of his life-giving Spirit, have thereby full power to strive against Satan, sin, the world, and their own flesh, and to win the victory; it being well understood that it is ever through the assisting grace of the Holy Ghost; and that Jesus Christ assists them through his Spirit in all temptations, extends to them his hand, and if only they are ready for the conflict, and desire his help, and are not inactive, keeps them from falling, so that they, by no craft or power of Satan, can be misled nor plucked out of Christ’s hands, according to the Word of Christ, John 10:28: “Neither shall any man pluck them out of my hand.” But whether they are capable, through negligence, of forsaking again the first beginning of their life in Christ, of again returning to this present evil world, of turning away from the holy doctrine which was delivered them, of losing a good conscience, of becoming devoid of grace, that must be more particularly determined out of the Holy Scripture, before we ourselves can teach it with the full persuasion of our mind.
I just can't determine with certainty whether it is possible for a born again believer to leave their salvation.
Here are some of the Scriptures supporting eternal security:
Romans 8:38-39 NASB For I am convinced that neither death, nor life, nor angels, nor principalities, nor things present, nor things to come, nor powers, (39) nor height, nor depth, nor any other created thing, will be able to separate us from the love of God, which is in Christ Jesus our Lord.
and
John 10:27-30 NASB "My sheep hear My voice, and I know them, and they follow Me; (28) and I give eternal life to them, and they will never perish; and no one will snatch them out of My hand. (29) "My Father, who has given them to Me, is greater than all; and no one is able to snatch them out of the Father's hand. (30) "I and the Father are one."
and
2 Timothy 1:12 NASB For this reason I also suffer these things, but I am not ashamed; for I know whom I have believed and I am convinced that He is able to guard what I have entrusted to Him until that day.
Here are some of the verses that support the possibility of final apostacy:
Hebrews 6:4-6 NASB For in the case of those who have once been enlightened and have tasted of the heavenly gift and have been made partakers of the Holy Spirit, (5) and have tasted the good word of God and the powers of the age to come, (6) and then have fallen away, it is impossible to renew them again to repentance, since they again crucify to themselves the Son of God and put Him to open shame.
and
John 15:6 NASB "If anyone does not abide in Me, he is thrown away as a branch and dries up; and they gather them, and cast them into the fire and they are burned.
and
Romans 11:19-22 NASB You will say then, "Branches were broken off so that I might be grafted in." (20) Quite right, they were broken off for their unbelief, but you stand by your faith. Do not be conceited, but fear; (21) for if God did not spare the natural branches, He will not spare you, either. (22) Behold then the kindness and severity of God; to those who fell, severity, but to you, God's kindness, if you continue in His kindness; otherwise you also will be cut off.