What was the name given to the trade system where finished manufactured goods were sent to africa in exchange for slaves who were sent to the americas and exchanged for colonial goods that would finally be returned to the home country?

You seem to be referring to the Atlantic triangular slave trade.  In this, the Americas sent Europe cotton, sugar, and tobacco.  Europe sent Africa manufactured goods, textiles, and rum.  Then, Africa sent slaves to the Americas, and the triangular trade continued.  You might want to flip through your textbook a little on this one: the exact name varies from the one I gave you.

the Atlantic triangular slave trade (check in your text, the name varies)

The Triangular Trade: triangle trade route brought African slaves to America.  In the next step, ships carried rum and guns to Africa. In Africa these were exchanged for slaves.