According to the doctrine of "popular sovereignty, or "squatter sovereignty," (ch. 13)
territories should extend the right to vote to all male settlers in the far west.
native indian peoples had a right to hold on to the lands they were already cultivating.
territories should have the right to expand their territory.
the united states should rule mexico directly.
residents of a territory should be allowed to decide whether or not to permit slavery.

"Residents of a territory should be allowed to decide whether or not to permit slavery" is the correct option.

Explanation:

Popular sovereignty was called Squatter Sovereignty in New England. It was a controversial political doctrine that allowed the federal territories to decide themselves if they want to enter the Union as Free state or Slave states. It was first used in 1850 while organizing Utah and New Mexico. Its first major implication was seen with the passage of Stephen A. Douglas's Kansas-Nebraska Act in 1854, it repealed the prohibition of slavery north of the 36 Degree 30 minutes.

du boi was considered a radical because he wanted the immediate end for segregation. the biggest different between him and booker t. washington is that, even though washington believed that segregation should be considered illegal, he did not believe at all that a law was needed to keep blacks away from whites and that this would be something natural (to have the races apart) for a long time until society adjusted, while du boi believe that agitation was the only way to end segregation.