Learn Turkish: Geçerli bir pasaport kullanabilir miyiz?

The root is kullanmak (to use). We add the ability vowel -a, then -bilir (can) to make kullanabilir (can use). For the "can we?" question, we add the question word mi as a separate word, a buffer letter -y-, and the "we" ending -iz, giving us kullanabilir miyiz.

Why is "bir" placed between "geçerli" and "pasaport"?

In Turkish, when a noun has both a descriptive adjective (like geçerli) and the indefinite article bir (a/an), bir is always placed right before the noun. So "a valid passport" translates literally as "valid a passport" (geçerli bir pasaport).

What is the root of the word "geçerli" (valid)?

It comes from the verb geçmek (to pass). The form geçer means "it passes", and adding the suffix -li (having/with) creates geçerli. So a document that is "valid" is literally described as something that "has the ability to pass".

Why doesn't "pasaport" have an accusative case ending here, since it is being used?

We only use the accusative case ending (which marks the direct object) for specific, definite objects, like when saying "the passport" or "my passport". Because we are asking about "a valid passport" in general (geçerli bir pasaport), it is indefinite and stays in its bare dictionary form.