zerschlagen
to smash
To break or shatter something into many pieces with great force.
Der Einbrecher hat die Fensterscheibe mit einem Stein zerschlagen.
The burglar smashed the windowpane with a stone.

to smash / break up
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to smash
To break or shatter something into many pieces with great force.
Der Einbrecher hat die Fensterscheibe mit einem Stein zerschlagen.
The burglar smashed the windowpane with a stone.

The prefix 'zer-' is inseparable and is never separated from the verb. It often expresses destruction or breaking apart, which can be seen in all meanings of 'zerschlagen' (to smash / break up).
Er hat aus Wut sein altes Handy zerschlagen.
Out of anger, he smashed his old phone.
In the physical meaning, 'zerschlagen' (to smash) is often used for fragile objects made of glass, ceramic, or porcelain. Typical objects are a bottle, a plate, or a windowpane.
Vorsicht, hier liegen Scherben, jemand hat eine Flasche zerschlagen.
Careful, there are shards here; someone smashed a bottle.
This meaning is metaphorical. Here, you do not 'zerschlagen' (smash) objects, but abstract structures. Typical objects are organizations such as a gang, a cartel, or a ring.
Der Polizei gelang es, den Schmugglerring zu zerschlagen.
The police managed to break up the smuggling ring.
For failed plans or hopes, the verb is used reflexively: 'sich zerschlagen' (to fall through). The subject is always the plan or the hope, not the person.
Mein Plan, am Wochenende wegzufahren, hat sich leider zerschlagen.
Unfortunately, my plan to go away for the weekend fell through.
to destroy
'zerstören' and 'zerschlagen' are close in meaning because both can express severe destruction or damage. 'zerstören' is more general and fits many kinds of damage, including damage to feelings, relationships, or plans. 'zerschlagen' often emphasizes that something breaks into pieces with force, for example glass or dishes. Figuratively, 'zerschlagen' is also used when an organization or a plan is permanently broken up or destroyed. Learners may confuse the words because both refer to an end or major damage. As a guide: 'zerschlagen' fits especially well with pieces, force, or a plan that has definitively failed; 'zerstören' is more general and focuses on the result.

Unsere Pläne für das Wochenende haben sich wegen der Arbeit zerschlagen.
Our plans for the weekend fell through because of work.
Ein falsches Gerücht kann eine gute Freundschaft zerstören.
A false rumor can destroy a good friendship.