Learn French: Je demandais une prolongation pour finir avec soin. - I was requesting an extension to finish with care.

This is the 'imparfait d'atténuation' (imperfect of politeness). By using the imperfect instead of the present tense (Je demande), the request sounds softer, less demanding, and much more polite. It is similar to saying 'I was wondering if I could get an extension' in English.

The translation says 'I was requesting,' but does that mean the action is strictly in the past?

Not necessarily. While it literally translates to a past action, native speakers frequently use this structure for a present request. If you walk into your boss's office today to ask for more time, you would use Je demandais to be polite about what you want right now.

Why is finir in the infinitive form here?

In French, when you use a preposition like pour (meaning 'to' or 'in order to') to show the purpose of an action, it must be followed by a verb in its unconjugated infinitive form. Here, pour finir expresses 'in order to finish'.

Could I use the adverb soigneusement instead of the phrase avec soin?

Yes, saying pour finir soigneusement (to finish carefully) is grammatically correct and means the exact same thing. However, using avec plus a noun is a very common and natural stylistic choice in French to describe how an action is performed.