How does the quote below from Romeo and Juliet demonstrate that hate can be caused by love and be overcome by it?”Here’s much to do with hate,...

This is a very famous quote from this excellent tragedy,
and, as you have identified, has a lot to do with the relationship between love and hate
as expressed in this play. This quote uses a series of paradoxes to point out the idiocy
of the situation that plagues the households of the Capulets and the
Montagues.

Let us remind ourselves of the context of this
scene. Benvolio has drawn near to Romeo to talk to him about his aloof behaviour and how
depressed he appears to be, and Romeo, trying to change to subject, remarks on the
"fray" that he obviously sees some form of evidence of and utters the quote that you
have cited. Note how he refers to the violence as having much to do with "hate," but
actually being evidence more of "love"--the love and pride that each family has in its
own identity. It is for love of their household that the members of the two households
feud, not hatred, which thus leads to the two paradoxes that Romeo uses: "O brawling
love, O loving hate." As for the second part of your question, I don't necessarily think
that these lines alone show that hate can be overcome by love. Rather I think they just
explain the paradox that lies at the heart of the violence of the play. To find evidence
of how hatred is overcome by love, look at how the deaths of Romeo and Juliet bring
peace to the feud.