You already know why the passive gets chosen: it puts the thing that receives the action first, when the doer is unknown, obvious, or not important (add by + the doer only when you want to name them). None of that changes here. What is new is that a whole passive can sit after a modal — can, should, must, might — to say whether the action is possible, a good idea, or necessary.
the same choice, now after a modal
This problem can be solved. → the focus is the problem, not who solves it.
The door must be locked. → what matters is the door, not the person.
This report can be checked by a manager. → add by + doer only when you want to name who.
Try it: Why choose "The door must be locked" instead of "Someone must lock the door"? (Because what matters is the door, not who locks it. / Because the door does the locking itself. / Because "lock" has no past participle.) Answer: Because what matters is the door, not who locks it. — The passive puts the receiver (the door) first when the doer isn't important or isn't known. Here we care about the door being locked, not about who does it — so: The door must be locked.
the past participle, unchanged after a modal
The past participle after be is built exactly as before: most verbs add -ed (solve → solved, lock → locked, check → checked), and the common irregular ones you learn one by one (do → done, send → sent, make → made, break → broken, write → written). Putting a modal in front changes nothing about it.
clean → cleaned: The room can be cleaned.
send → sent (irregular): The form should be sent.
do → done (irregular): This must be done.
Try it: This report must be ___ today. (write → past participle) (written / writed / write) Answer: written — "Write" is irregular, so its past participle is written, not "writed." After be you need the past participle, not the base verb write: The report must be written today.
a modal + be + the past participle
To put a passive after a modal (can, should, must, might), use modal + be + the past participle. The be stays in its plain form after the modal.
We can solve this problem. → This problem can be solved.
You should finish the work today. → The work should be finished today.
Someone must lock the door. → The door must be locked.
Try it: We can solve this problem. Turn this around so the problem comes first. (Leave out the word you don't need.) Answer: This problem can be solved. — After the modal comes be + the past participle: This problem can be solved. "Solve" is the base verb, but the passive needs the past participle solved after be, so leave it out.
keep be, don't change it
After a modal, be never changes to is, are, or was. It always stays as be.
You must do it by Friday. → It must be done by Friday.
Sign the form here. → The form should be signed here.
They might delay this. → This might be delayed.
Not "It should is finished." — say "It should be finished.". After a modal, use be, never is or was: should be, not "should is".
Try it: Someone should send the form today. Build the passive. The verb be stays plain. (Leave out the word you don't need.) Answer: The form should be sent today. — After the modal, be stays plain: The form should be sent today. "Is" can't follow a modal (not "should is"), so leave it out, it's modal + be + the past participle.
choosing can / should / must / might
The be + past participle part stays the same — the modal you choose changes the meaning: can = it's possible / able to happen; should = it's a good idea / recommended; must = it's necessary / required (no choice); might = it's possible but not sure (maybe).
This bottle can be recycled. → it's possible.
This answer should be checked. → it's a good idea.
This form must be signed. → it's required, no choice.
The train might be delayed. → maybe, we're not sure.
Try it: The rules say every visitor is required to sign this form. There is no choice — signing it is compulsory. Which passive fits "it's required, no choice"? (This form must be signed. / This form should be signed.) Answer: This form must be signed. — "Required, no choice" is must: This form must be signed. Should would only make it a good idea — a recommendation you could ignore — which does not match "compulsory."
making it negative
To make the passive negative, make the modal negative and keep be + the past participle: can → can't be done, should → shouldn't be done, must → mustn't be done.
This can't be solved. → it's impossible.
This shouldn't be left open. → it's a bad idea.
This mustn't be touched. → it's not allowed.
Not "This can't do." — say "This can't be done.". The negative still needs be + the past participle: can't be done, not "can't do".
Try it: Nobody can do this. Build the negative: it's impossible to do this. (Leave out the word you don't need.) Answer: This can't be done. — The negative modal still takes be + the past participle: This can't be done. "Do" is the base verb, but after be the passive needs the past participle done, so leave it out.
asking a passive question
To make a passive question, put the modal first: modal + subject + be + past participle? The be + past participle part doesn't change — only the modal moves to the front.
This can be fixed. → Can this be fixed?
The form should be sent. → Should the form be sent?
It must be finished today. → Must it be finished today?
Try it: This can be fixed. Make it a question: ask if it's possible. (Leave out the word you don't need.) Answer: Can this be fixed? — For a passive question, the modal goes first and be + the past participle stay together: Can this be fixed? You don't add "does" — the modal can already makes the question, so leave it out.