zerstreuen
to disperse
To drive apart or distribute a group of people or a collection of things in different directions.
Die Polizei zerstreute die Menschenmenge mit Wasserwerfern.
The police dispersed the crowd with water cannons.

to disperse / dispel / distract
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to disperse
To drive apart or distribute a group of people or a collection of things in different directions.
Die Polizei zerstreute die Menschenmenge mit Wasserwerfern.
The police dispersed the crowd with water cannons.

'Zerstreuen' (to disperse / dispel / distract) is an inseparable verb. The prefix 'zer-' remains firmly attached to the verb stem 'streuen' (to scatter) in all conjugated forms and sentence types.
Der Wind hat die Papiere auf meinem Schreibtisch zerstreut.
The wind scattered the papers on my desk.
The prefix 'zer-' often expresses separation, dissolution, or movement in many different directions. It intensifies the base verb 'streuen' (to scatter) into a complete distribution or dissolution.
Nach dem Konzert zerstreute sich die Menge langsam.
After the concert, the crowd slowly dispersed.
In the meaning 'auseinandertreiben' (to drive apart), the verb is mostly used with objects that refer to a group. Typical phrases are 'eine Menschenmenge zerstreuen' (to disperse a crowd), 'Demonstranten zerstreuen' (to disperse demonstrators), or 'eine Versammlung zerstreuen' (to disperse an assembly).
Die Sicherheitskräfte versuchten, die Demonstranten friedlich zu zerstreuen.
The security forces tried to disperse the demonstrators peacefully.
To express that someone distracts themselves for relaxation, the verb is used reflexively: 'sich zerstreuen' (to divert oneself). The person distracting themselves is the subject of the sentence.
Am Wochenende fahre ich ans Meer, um mich ein wenig zu zerstreuen.
At the weekend, I'm going to the sea to divert myself a little.
This meaning is used almost exclusively with abstract nouns that express a negative emotion or uncertainty. Common combinations are 'Zweifel zerstreuen' (to dispel doubts), 'Bedenken zerstreuen' (to dispel concerns), or 'Sorgen zerstreuen' (to dispel worries).
Ihre Erklärung konnte meine letzten Zweifel zerstreuen.
Her explanation was able to dispel my last doubts.
In physics, 'zerstreuen' (to scatter) is a technical term. It describes how radiation, such as light, is deflected by a medium in different directions. This use is found in scientific texts.
Die Atmosphäre zerstreut das blaue Licht der Sonne stärker als das rote.
The atmosphere scatters the sun's blue light more strongly than the red.
to distribute / spread
'verteilen' and 'zerstreuen' can both mean that people or things are no longer together in one place. 'verteilen' usually sounds neutral or planned: people position themselves in different places, or things are deliberately divided up. 'zerstreuen' sounds more as if a group is being driven apart or breaking up in different directions. Learners may confuse the words because both have to do with moving apart. 'verteilen' fits better when there is order and planning, while 'zerstreuen' fits more with a crowd or group that breaks up or is driven away.

Nach der Durchsage zerstreute sich die Menge vor dem Eingang schnell.
After the announcement, the crowd in front of the entrance quickly dispersed.
In der Schule haben sich die Kinder auf dem ganzen Hof verteilt.
At school, the children spread out across the whole playground.