Verb Patterns

No the second sentence is not correct.

The verb is "to be", which is the infinitive form. This use of the verb believe takes the infinitive form.

What does it imply? Could you be more specific? I stil don't understand it too much.

why then, when it comes to "I admit to being strict with her children. There is also the verb "to be".
Thanks!

They are two totally different forms. They both happen to use the word "to". Don't let that confuse you. They are completely different forms.

In the first sentence, "to be" is being used as a present infinitive. You can think of this as one word really, and in many languages (e.g. Spanish) the infinitive is one word.

In the second sentence the word "to" is a preposition. It is connecting "admit" with "being strict".

It is just a coincidence that the same word is used in these two different ways.

Thanks a lot!! And when it comes to this, why in the gap there is not "offered"?
DESPITE
David was still dubious about accepting the job ......................him a higher salary.

The correct answer: "despite the company offering"