"Who" versus "whom" with multiple verbs

Solution 1:

In this instance, the pronoun "who" is the object of the verb "know". So you want to use objective case whom.

Solution 2:

The quantity of verbs has no effect on the choice between who and whom. The only thing you need to figure out is whether or not you need a subject for a verb.

If who/whom is the subject of a verb, use who. Otherwise, use whom.

The non-technical instruction on choosing the right word: Who can only be used as a subject, so if you don’t need a subject, don’t use who; use whom.

For this example, the correct choice is whom. “The man who/whom I know to be unhappy…” I suppose this is followed by a verb. The simple subject of the sentence (of the verb that follows) is man, not who/whom. Who/whom is not the subject of a verb, so you use whom.

Another way to write the example, which might make it easier to parse, is as follows: “The man, whom I know is unhappy,…” As we see, whom is not serving as the subject of any verb.

Source: Precise Edit

Solution 3:

As I write, all other answers agree that whom is the correct choice in this construction (because "whom I know to be unhappy" is an auxiliary phrase, wherein whom is not the subject of a verb).

I don't dispute the strict grammatical position, but I would say that, as suggested by this NGram, whom appears to be increasingly falling into disuse.

Correspondingly, here are over 1000 written instances of "who I know to", most if not all of which are "incorrect" according to strict grammar. In my opinion, whom is already becoming somewhat 'dated', and it's only a matter of time before it disappears completely.

Solution 4:

As the question is tagged with acceptability, I will report the following paragraph, reported by NOAD in the usage of who section:

The normal practice in modern English is to use who instead of whom (Who do you think we should support?) and, where applicable, to put the preposition at the end of the sentence (Who do you wish to speak to?). Such uses are today broadly accepted in standard English, but in formal writing it is best to maintain the distinction.

If you want to avoid writing who when you should use whom (or vice versa), you can use that.

the man that I know to be unhappy

That is a relative pronoun used to introduce a defining or restrictive clause, especially one essential to identification; it is used instead of when, which, who, whom.

the book that I have bought yesterday
the person that I will meet tomorrow
the year that Anna was born

Solution 5:

"I know the man to be unhappy" seems be more informative, but it doesn't answer the question until you change it again to "Him I know to be unhappy". Or how about "I know that man [him] to be unhappy. This would indicate that you want the objective case - whom.