Part of the problem is that there are probably over a billion Muslim people in the world, yet we only hear about a minority that are terroristic. When I do a "back of the envelope" calculation, I've decided that probably less than 1 Muslim person in 500 has anything to do with terrorism and/or extremism.
We have Islam to thank for much of our Western culture.
All of those stories about Hercules, Athena, Zeus, Apollo, and so forth? The history of Greece and Rome?
We have Islam to thank for this continuity with our history. Muslim scholars preserved this culture during the dark ages of the Middle Ages . . . before they gave it back to us during the Enlightenment.
Algebra (and other forms of advanced mathematics) exist because Islam gave us our number system . . . which they borrowed from India, refined, and gave to the West.
They shared their medical discoveries with the West (Rhazes was the greatest physician of his time, and he introduced the ideas that cleanliness helps prevent disease, and that a patient's progress under a doctor's care should be charted, and made available to other doctors so that patient care mistakes need not be repeated), their profound discoveries in astronomy, and also started the discipline of sociology.
I could go on, but I'm sure you get the idea.
To judge all of Islam by the actions of a subgroup of violent assholes is like judging all of Christianity by the standards of the Ku Klux Klan.