Learn Swedish: Polisen måste ha utrett mordet. - The police must have investigated the murder.

It means we are making a strong deduction: we assume they did. Just like in English, using måste ha followed by a supinum (utrett) expresses a logical conclusion about the past. If you wanted to say they were required to investigate it in the past, you would drop ha and use a past-tense obligation, like Polisen var tvungna att utreda mordet (The police had to investigate the murder).

Why is the verb form utrett here, and what is its infinitive?

The infinitive is utreda (to investigate). Because it follows the auxiliary verb ha, it must be in the supinum form. Utreda is based on the verb reda, which is a Group 3 verb (reda, redde, rett). Therefore, the supinum of utreda is utrett.

In English, 'the police' is grammatically plural. Is polisen plural in this sentence?

No, polisen is singular. In Swedish, polisen usually refers to the police force as a singular entity (or a single police officer). While it doesn't change the form of the modal verb måste here, it is an important difference to remember for adjective and pronoun agreement. If you need to explicitly refer to multiple officers, you would use the plural poliserna.

I've seen the phrase reda ut before. Can I use that instead of here?