I want a regular expression to check that:
A password contains at least eight characters, including at least one number and includes both lower and uppercase letter
Regular expressions don't have an AND operator, so it's pretty hard to write a regex that matches valid passwords, when validity is defined by something AND something else AND something else...
But, regular expressions do have an OR operator, so just apply DeMorgan's theorem, and write a regex that matches invalid passwords:
Anything with less than eight characters OR anything with no numbers OR anything with no uppercase OR or anything with no lowercase OR anything with no special characters.
So:
^(.{0,7}|[^0-9]*|[^A-Z]*|[^a-z]*|[a-zA-Z0-9]*)$
If anything matches that, then it's an invalid password.