If the following molecule is considered and subjected to bromination, then at how many positions bromination process forms a major product?

Bromination proceeds with free radical mechanism. Where there is a possibility of the formation of a tertiary radical, there is huge chances of bromination. Among the given compound, in all tertiary carbon (associated with at least one H atom) undergoes bromination which implies that Br can attach at the respective position.

Bromination always prefer tertiary position, as per IUPAC we can see that 4^th position is tertiary.