zbadać coś dogłębnie in English:

The police have promised to look into the problem.

Someone said there was a meeting at 9:30 but I haven't heard anything about it. Shall I look into it?

[We will look into the problem as soon as possible.]

I'll look into the matter and send an invoice.

We'll look into the problem and come back to you when we have the information.

I wrote a letter of complaint, and the airline have promised to look into the matter.

Don't worry, I'll look into it.

We'll have to look into the problem carefully.

He will look into your complain.

We must look into the matter before we make a decision.