面倒: troublesome tasks and caring for someone
Read 面倒 through a little troublesome paperwork and a friend caring for a dog. The constructions 面倒だ and 面倒を見る distinguish an evaluation of a task from an act of care.
Paperwork described as a little troublesome
「複雑な書類を書くのは少し面倒だ。」 (Filling out complicated paperwork is a little troublesome.)
The evaluation concerns filling out complicated paperwork. 少し keeps the degree at ‘a little’; the sentence does not report abandoning the task.
The friend who cared for the dog
「旅行中、友人が犬の面倒を見てくれた。」 (My friend took care of my dog while I was traveling.)
Here 面倒を見る describes providing care. The friend is the caregiver, and the dog is the one cared for during the trip.
Reading the construction as well as the noun
面倒だ evaluates a task as troublesome, whereas 面倒を見る describes caring for someone. The dog's care is not automatically being evaluated as unpleasant because the same noun appears.
One sentence evaluates paperwork, and the other reports the friend's care. Their objects and actions distinguish the two uses of 面倒.
Does 犬の面倒を見てくれた mean the friend merely looked at the dog and found it troublesome?
Show the answer and why
No. It says the friend took care of the dog. Read the care construction 面倒を見る separately from the evaluation expressed by 面倒だ.