Learn English – passive voice of ‘I have a pen to write’

Break into is a transitive verb. This is what the Merriam-Webster Unabridged says. Other dictionaries seem to differ, but in this answer the verb will be used in a transitive sense.

Specifically, it is a prepositional verb because the verb + preposition make one meaning-unit. (A prepositional verb is not a phrasal verb, even if some dictionaries wrongly list break into with phrasal verbs. (There are tests to distinguish the two types of verbs; I don't want to overlong this answer with them, but I can provide).

Without an object, the sentence is incomplete and thus ungrammatical:

ungrammatical without an object:
*He broke into.

grammatical forms:

active:
He broke into the house

passive:
The house was broken into by him.

Note the sense of action or motion or direction (He broke into the house).

Notice that break into cannot always be used in the passive. It is precisely when the verb does not have a sense of motion/action that this is the case.

He broke into a sweat.

*A sweat was broken into by him.

The audience broke into applause.

*Applause was broken into by the audience.

*A gallop was broken into by the horse

Last, another transitive verb, composed of one word, can substutute for break into: burgle. (Note there are no exact synonyms and let's not get into legal definitions.)

This is a silly question. It asks one to do something that can't be done.
The sentence

• I must go and look for my brothers.

is Intransitive, but Passive can only apply to a Transitive clause.

The Passive rule promotes the direct object to subject, demotes the old subject to an optional object of by, and adds the auxiliary verb be before the past participle form of the main transitive verb. Like this:

• Marie shot my cousin. == Passive => My cousin was shot (by Marie).

But if the clause isn't transitive, there isn't a direct object to promote, and other noun phrases don't usually work.

• Mary slept all day. but not *All day was slept (by Mary).

Sometimes prepositional objects can be passivized, if the preposition is one that makes a transitive verb out of an intransitive one, like look (at) or listen (to)

• We must look at/listen to that again. == Passive => That must be looked at/listened to again.

But that's rarely the case, so most prepositions after verbs don't mark direct objects. And without a direct object, Passive is impossible.

The real solution is to get a new textbook that actually describes English, instead of something like English.