Learn French: L'aluminium doit être broyé, mais il ne peut pas être incinéré. - Aluminum must be shredded, but it cannot be incinerated.

It maps perfectly to English. You conjugate the modal verb (doit or peut), add the infinitive être (to be), and finish with the past participle of the main verb (broyé or incinéré).

Why does the negation pas go after peut instead of after être?

When negating a conjugated verb that is followed by an infinitive, the ne ... pas sandwich wraps around the conjugated verb. Since peut is the conjugated verb, we write ne peut pas, leaving the infinitive phrase être incinéré for afterward.

Do the past participles broyé and incinéré have to agree with the subject, even after an infinitive?

Yes, they do! Because they describe the subject l'aluminium in a passive construction, they must agree in gender and number. Here, they are masculine singular. If the subject were feminine, like la canette (the can), they would change to broyée and incinérée.

Is it common to use the passive voice like this in everyday spoken French?

While perfectly correct, the passive voice often sounds formal or technical. In everyday spoken French, you would more likely use the active voice with on, saying: On doit broyer l'aluminium, mais on ne peut pas l'incinérer.