How did the belief in manifest destiny and the issue of oregon lead to the annexation of texas?

Manifest Destiny was the belief that it was the United States destiny to expand throughout the entire the continent. It set the tone for a great deal of expansion of the US territory including Texas and Oregon. 

Manifest destiny played a really important role in the expansion to Oregon, which was in dispute with Britain. Thousands of Americans migrated to Oregon in the 1840s over the Oregon Trail. Britain was trying to move the boundary line to include more of present day Washington state but people in support of Manifest Destiny wanted all of Oregon. There was a lot of back and forth on the issue, especially when President Polk was elected. Eventually they settled with only part of Oregon (at the time) because the US was able to enter the Mexican-American War and did not want to fight with Great Britain as well. 

Texas was originally owned by Mexico but in 1836 they declared their independence from Mexico and became the Republic of Texas. Then Texas requested to join the US. While the US wanted the land, it was controversial because Texas was place where slaves could be owned and a lot of Democratic politicians in the US government disagreed on the subject. James K. Polk favored the annexation of Texas and when he was elected as president he approved expansion to Texas. 

Essentially, Manifest Destiny acted as a platform for people to support expansion to places like Texas and Oregon despite the conflicts they encountered in acquiring the lands.