Ok, here is the way I read it.
Romans 8:28-39
28And we know that in all things God works for the good of those who love him, who have been called according to his purpose. 29For those God foreknew he also predestined to be conformed to the likeness of his Son, that he might be the firstborn among many brothers. 30And those he predestined, he also called; those he called, he also justified; those he justified, he also glorified. 31What, then, shall we say in response to this? If God is for us, who can be against us? 32He who did not spare his own Son, but gave him up for us allhow will he not also, along with him, graciously give us all things? 33Who will bring any charge against those whom God has chosen? It is God who justifies. 34Who is he that condemns? Christ Jesus, who diedmore than that, who was raised to lifeis at the right hand of God and is also interceding for us. 35Who shall separate us from the love of Christ? Shall trouble or hardship or persecution or famine or nakedness or danger or sword? 36As it is written:
"For your sake we face death all day long;
we are considered as sheep to be slaughtered." 37No, in all these things we are more than conquerors through him who loved us. 38For I am convinced that neither death nor life, neither angels nor demons, neither the present nor the future, nor any powers, 39neither height nor depth, nor anything else in all creation, will be able to separate us from the love of God that is in Christ Jesus our Lord. (NKJV, Highlight mine)
The words translated predestined in the above verses are from the Greek word proorizo, which carries the meaning of determine beforehand, ordain, to decide upon ahead of time. So, predestination is God determining certain things to occur ahead of time. What did God determine ahead of time? According to Romans 8:29-30, God predetermined that certain folks would be conformed to the likeness of His Son, be called, justified, and glorified. Essentially, God predetermines that certain individuals will be saved. There are numerous scripture verses that refer to believers in Christ being chosen like Matthew 24:22 and 31, Titus 1:1, and 1 Peter 1:1-2 among others. Predestination is biblical doctrine but in the sense that God has chosen certain individuals to be saved.
Now, If God is choosing who is saved then how does that compare with biblical passages that show that we can be saved if we chose and believe in Christ? The Bible says that we have the choice and that all who believe in Jesus Christ will be saved like in John 3:16 and Romans 10:9-10 among others. The Bible never describes God rejecting anyone who believes in Him or turning away anyone who is seeking Him. The only possible answer is that God both predestines and allows those that believe in Christ to be saved. God predestines who will be saved, and we must choose Christ in order to be saved. Both facts are equally true.