There are German verbs with a special feature, that does not exist in English and many other languages. It's called:
trennbare Verben (separable verbs)
A separable verb is a verb that can be split into two separate words, and the extra bonus is, that those parts can merge again to one word, enclosing some interfixes, that otherwise would be a distinct word (»zu«) or a prefix (»ge«).
»Aufhören« (to stop) is such a verb:
I werde mit dem Rauchen aufhören.
I will stop smoking.
This example was »Futur I« where you use the auxiliary verb »werden« plus the infinitive. So in this tense, you use the word as it is written in a dictionary.
But when you use a separable verb in present tense, it looks this way:
Ich höre mit dem Rauchen auf.
I stop smoking.
See? The word was split into its parts, and the former prefix moved to the very end of the sentence. This also can be true for longer sentences:
Ich höre, wie ich dir bei deiner Geburtstagsfeier, bei der auch Tante Agnes und Onkel Hannes waren, versprochen habe, hier und heute mit dem Rauchen auf.
Past tense is build using an auxiliary verb (a form of haben or sein) plus a form of the verb that is built by adding the prefix »ge«:
Ich schlafe. Ich fahre.
Ich habe geschafen. Ich bin gefahren.
If the verb is a separable verb, you do it this way:
Ich habe mit dem Rauchen aufgehört.
I stopped smoking.
Did you see how the syllable »ge« now squishes in between the two parts of the separable word? It's not a prefix here, it's an interfix.
And now comes the magic thing: zu + Infinitive. If you use non-separable verbs, then »zu« is a word and the infinitive is another word:
Ich habe Lust zu schlafen.
I'd like to sleep.
And this is what happens with separable verbs:
Ich habe Lust mit dem Rauchen aufzuhören.
I'd like to stop smoking.
So when ever you find »zu« or »ge« as second syllable in a verb, take this as a very strong hint that you are looking at a separable verb. Delete this interfix, and you get the infinitive that can be found in a dictionary.