grammar - Feel/believe someone to be adjective/noun VS feel/believe someone adjective/noun

1. The company believes their software to be useful for Internet users.



2. The company believes their software useful for Internet users.


3. Do you feel beauty to be important in getting a job?


4. Do you feel beauty important in getting a job?

Which numbers are correct?

And any reason why you didn't choose the other numbers ?

And what are the differences in meaning?

Some native speakers I asked about them argue that they're all unnatural.