In your teaching from Hebrews, you say that a believer's failure will be judged by God. But the Bible says that God will "remember our sins no more" and that our sins are separated from us as far as the East is from the West? How can the Gospel be Good News if God is remembering our sins?
You are conflating two different judgments. The Christian is forgiven of sin by the blood of Christ, so our sins are not "remembered" by God in the sense that He will not hold us accountable for them with the penalty of the Second Death. This is the Great White Throne Judgement and believers will not face this judgment:
However, there is another judgment for the believer in which Jesus determines our eternal rewards:
This is the Judgment Seat of Christ, which takes place at the resurrection (i.e., Rapture) of the Church. This judgment is for the basis of rewards, and as Paul says, it takes into account those things we've done, whether good or bad. A Christian who lives a life of disobedience will see eternal consequences in the form of a loss of eternal reward. At that time, all our works, good and bad, are in view for that evaluation, Paul says:
Once again, this is not a judgment for determining salvation, for salvation is determined by faith alone and not by works. But our works are evaluated for eternal reward, and that is the concern of Hebrews 4. So the GOSPEL is very good news indeed for all who come to faith in Jesus Christ, and now that we are called by His name, we are to serve our Master obediently. Failure to serve the God Who saved you is not good as the writer of Hebrews says later:
That passage was written to believers (notice in v.31 the writer says God will judge "His" people). The modern church has done the believer a great disservice by downplaying or ignoring teaching on the reality of a judgment for the believer, which has allowed generations of believers to remain ignorant of their obligations to serve Christ. As Peter says, the believer will be the first to be judged:
For more background on eternal rewards and the judgment believers will face, please read our article on Eternal Rewards.