Learn Farsi: ضمانتی که آن را تمدید کردی برای این دستگاه است. - The guarantee that you extended is for this device.

This is a key difference between Farsi and English! In Farsi, when the noun you are describing (the guarantee) is the direct object of the verb inside the "that" clause (you extended it), you must include a "resumptive pronoun" to refer back to it. Here, ân means "it" and râ marks it as the specific object of tamdid kardi (you extended). So literally, Farsi says: "The guarantee that you extended it..."

Why does zemânat (guarantee) have the -i suffix (zemânati) if the translation says "the" guarantee instead of "a" guarantee?

While the -i suffix often means "a" or "an", here it is acting as the restrictive -i. When a noun is followed by a ke (that/which) clause that specifies exactly which one we mean, we attach -i to the noun. Because it makes the noun specific, we translate it into English using "the".

Could I use an object ending instead of ân râ to say "it"?

Yes! Instead of using the separate words ân râ, you could attach the "it" ending -ash to the non-verbal part of the compound verb. The sentence would become: zemânati ke tamdidash kardi barâye in dastgâh ast. Both are perfectly correct and mean the exact same thing, though the attached ending is very common in everyday spoken Farsi.

What kind of verb is tamdid kardi?

It is a compound verb made of (extension or renewal) and (to do). Together, means "to extend" or "to renew" (like a contract, visa, or guarantee). Here it is conjugated in the simple past for "you" (singular): .