Which of the five founding ideals were the Radical Republicans trying to extend to the Freedmen and through what
programs?

The five founding ideals: Rights, Opportunity, Liberty, Equality and Democracy

The programs: Freedmen's Bureau

Explanation:

The "radical republicans" refers to a group of people who wanted to punished the southern states for ignoring Lincoln's order to abolish slavery. Most of the people in this group wanted to treat African American people as equal citizens.

Even after the salves became free following the civil war, the Freed men still struggling to fulfill their basic needs. They're still heavily discriminated against.

The radical republicans wanted the freed slaves to obtain  protection to have their own Rights, Opportunity, Liberty, Equality and Democracy. The ideals that were protected to all untied states citizens within the constitution.

In order to help the freed slaves achieve this, they established the Freedmen's bureau. This organization was created to provide wide variety of helps for the Freed slaves in order to survive the transition from slavery to legal citizens. The bureau provide them with things such as food, shelter, education, legal defense, etc.

c) legal battles last five weeks after the election due to the narrow victory in several states.