If you entered the country using someone else’s passport, you are inadmissible, and hence cannot receive permanent residency even if you are married to a US citizen. However, if your US citizen or a permanent resident spouse or parent needs your help, the I-601 extreme hardship waiver might be your only option.
The first question anyone should ask is what constitutes an extreme hardship in the eyes of the USCIS. The answer is not so simple. The immigration officer will look at all the details of your application before making a decision. Every answer counts, which means you need to eliminate the possibility of a misinterpretation. The extreme hardship can be demonstrated through an extreme financial hardship, medical hardship, psychological hardship, etc.