Final answer:
The Convention of 1845 concerned Texas's integration into the United States with implications for slavery. The main offers were about Texas entering as a slave state and handling its debts, along with more rigorous slave laws, reflecting the mounting national conflict over slavery.
Step-by-step explanation:
The two competing offers at the Convention of 1845 concerning Texas involved the annexation of Texas into the United States with various implications for slavery. One proposal was evidently to recognize and integrate Texas, a slave state, expanding potential territories allowing slavery and influencing the political balance between free and slave states. The other proposal involved more stringent conditions such as the payment of Texas's outstanding debts from the Lone Star Republic days and the end of the slave trade (though not the abolition of slavery) in Washington D.C., along with a more robust federal fugitive slave law as part of the Compromise of 1850.
Zachary Taylor, the U.S. President at the time, thought the Mexican Cession lands were too arid for cotton farming and hence would not attract slaveholders, while others argued for the right to move to these territories with slaves. Meanwhile, Texas's admission to the Union was understood by many as a clear move toward the extension of slavery, which raised concerns among antislavery advocates that the federal government was backing pro-slavery policies. The debates at the convention were indicative of broader national conflicts leading up to the American Civil War.