Learn Portuguese (Portugal): Eu gosto das tuas opiniões.

The verb gostar (to like) must always be followed by the preposition de (of). Because we are talking about feminine plural opinions, we need the article as (the). When you combine de and as, they contract to make das.

If das contains the word for "the", why are we saying "the" before "your"?

In European Portuguese, you usually must place a definite article (like o, a, os, or as) right before a possessive word. Combined with the de from gostar, the sentence literally translates to "I like of the your opinions".

How do we know to use tuas instead of teus?

Possessive words have to match the gender and number of the thing being owned, not the person who owns it. Since the singular word opinião is feminine, its plural opiniões requires the feminine plural word for "your", which is tuas.