How do I remove a 38 year old bankruptcy that appears in news archives section under Bankruptcy Business Records?

I doubt you can remove it. Such records are intended to be historical, and as accurate as their sponsors can make them. You could comfort yourself knowing that any discerning reader will see that the bankruptcy occurred long long ago and has no bearing on your present financial situation. Also, you can append to your credit reports from each of the three major credit reporting entities a statement, which I think can be up to 100 words, in which you explain anything you think needs explaining.

In the US, you cannot remove news records from a Google search. Laws in Europe address this concern, but I know of nothing that allows links to be removed. A better option would be to post so much new information about yourself that the old record gets buried. I have a former client that posts a picture of her dog with the weather report every morning, and this puts any old information about her on page 200.