Does DSB-SC signals retain original shape of the message signal \$ x(t) \$ when \$ x(t) <0 \$?
In non-suppressed carrier AM systems (typically AM broadcast transmitters), the envelope of the modulated carrier always retains the shape of the modulating signal and so it is easy for cheap AM radios to make a decent job of demodulating the radio signal successfully and with fairly low distortion on a budget.

With DSB suppressed carrier you are quite correct - the envelope detector will introduce huge amounts of distortion when trying to convert the radio signal to base-band.


  Then how do DSB-SC systems deal with modulating signals where x(t)&lt;0


This requires a specialized receiver and is used for specialized applications. Demodulation is done by typically extracting the un-modulated carrier signal by special means and then multiplying that extracted carrier with the radio signal. This converts the signal back to base-band (after appropriate filtering).

The special means for extracting the un-modulated carrier is typified by using a signal squarer (i.e. modulated signal is multiplied by itself) and this produces a coherent carrier at twice the actual carrier frequency of the radio signal. This can be fed via a gain-control to a divide-by-two circuit (similar to a D type flip-flop) and hey presto, the output is a carrier of the right frequency and no phase reversals each time x(t) drops through 0 volts.



Picture from here