"मैं इस इलाके में रहती हूँ।"
अनुवाद:I live in the region.
I live in these regions = "मैं इन इलाकों में रहती हूँ।"
I live in this region = "मैं इस इलाके में रहती हूँ।"
Hence, I live in this regions = "मैं इस इलाकों में रहती हूँ।"
This = इस , These = इन
Peoples by mistake use 'इस' and 'इन' as if these two things are same, but they are different, they are singular and plural form of same pronoun.
Also the thing is that, 'I live in this regions' is wrong as number of 'this' should be in agreement with number of 'regions', correct usage should be 'I live in these regions' .
If (let us assume, because there is not reason that it cannot happen) in a given context I live in this regions is valid, then मैं इस इलाकों में रहती हूँ। is also valid for that case.
P.S. If it hadn't been regions in the statement I would not have said anything. My focus was on 'regions' not 'the' .