Vacuuming. A simple household chore.

I thought what if you could attach a magnet to the motor which turns the broom around. Then encase the magnet with wiring to generate electricity. Would the generator be able to power itself (The broom motor) plus extra electricity?

Of course not. Because then you would have perpetual motion and free energy.

I can follow your pattern of thoughts here, but I also have to say no.

The extra weight that the magnets add requires that you increase the power of the engine. Already there you have the problem.

Sorry to wreck it for ya- but it's back to the drawing table for you...