They don’t care what he does because they regard who he is as beyond the pale.
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They don’t care what he does - the main clause in which the subordinate clause what he does is a direct object of the predicator don’t care;
because they regard who he is as beyond the pale - a subordinate clause (adverbial clause of reason) in which its subordinate clause who he is is a direct object of the verb regard and the degree (comparative?) adverbial as beyond the pale (with compound preposition as beyond) modifies the clause they regard who he is.
Is my analysis acceptable?
They don’t [care what he does because they regard who he is as beyond the pale]. Some people would treat “ they don’t care what he does ” as a main clause, but on my analysis it's only part of a clause. The predicator is just "care".
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They don’t [care what he does because they regard who he is as beyond the pale].
Some people would treat “they don’t care what he does” as a main clause, but on my analysis it's only part of a clause.
The predicator is just "care".
What he does is not an object, but a subordinate interrogative clause (embedded question) serving as complement of “care”.
tkacka15what he does is a direct object
Note that the verb "care" is intransitive only, so it never has a direct object.