What tool from the Compromise of 1850 quieted sectional divisiveness?
Given that the Civil War started up a mere 11 years after this compromise, it is clear that nothing in this compromise really did away with sectional enmity. However, you could argue that the tool of "popular sovereignty" did reduce sectional tensions for a time.
One piece of the Compromise of 1850 was the idea of popular sovereignty. It stated that the territories that would be made out of the Mexican Cession would be able to decide for themselves whether they wanted to be slave or free. This was in contrast to the territories made from the Louisiana Purchase which were assigned to be slave or free by the Missouri Compromise.
By doing this, the Compromise of 1850 at least put off sectional divisiveness until a later date. It made it so that Congress did not have to go through the rancorous process of deciding which areas would and would not be slave. However, just deferring a problem is not the same as solving it. This is why the Compromise of 1850 did not actually manage to fix the sectional problems.