What do you think they mean by "delicate, behind the scenes negotiations"

"Delicate, behind the scenes negotiations" means negotiations that are done in secret or in private between highly influential parties. Their nature is delicate because the situation is usually critical and important, hence why they might also require some level of privacy.

the passage of the Compromise of 1850

Explanation:

The above passage is an extract from a speech made by Stephen Douglas, October 1850 on account of the Compromise of 1850.

After the Mexican-American War, US established the Compromise of 1850 to resolve the disputes arising from the newly formed states. There were numerous discontentment amongst the people regarding the bills of the compromise.

The objectives of the Compromise of 1850 faced many criticisms and is believed to have been a major factor that led to the American Civil War.