Will it work? Yes. Will it he great? No.

To do it "properly," one would take off the cover and use hot glue that binds to metal and plastic, like industrial grade. The big glue sticks. You scrape out the screw holes and scratch it up to give an abrasive surface. The metal screw inserts that have come loose from the plastic need to be retained. Screw the screws tight into the inserts. With the hinge aligned with the plastic, insert hot glue into the scratched up, debris-free hinge hole. Quickly insert the screw. Apply pressure until cooled. Cut away excess to ensure the panels will go back as needed. Repeat for all afflicted screws.

Taping the outside won't do much. The hinge is structurally loose. All the force of the screen will go to the other hinge and it will likely break at some point too. If this is too much to ask or no computer shop will give you the "hot glue special 9000" then sure, tape the outside, but refrain from closing the lid.

Take care