finding out the bandwidth of m(t)*m(t) if the bandwidth of m(t) is a known quantity
Your bandwidth will double. You can derive this using trig identities:



DC and 2*theta components will form unless you filter them out.
I ran a quick simulation in MATLAB to confirm. F(m(t)) on the top-left, F(m(t)*m(t)) on top-right. Bottom-left is the the top-left zoomed in, and vice verse.



conclusions:


m(t) has a passband from 4kHz to 6kHz
m(t)*m(t) has twice the bandwidth, and massive DC componets


Side note, not related to your question:

This property is actually exploited in QPSK demodulation. Here the double frequency is eliminated almost perfectly by mathematically performing something called "integrate-and-dump". (Or you can always just use a low pass filter too). Any good communications textbook will describe this in detail.