The correct option is
A carefully
'Adverb' is a word that is used to change or qualify the meaning of an adjective, a verb, a clause or another adverb.In the given sentence, the verb is 'injured' and 'serious' is the level / details of his injury.
Option B - 'carefully' is an adverb which describes the verb 'injured'.
On the other hand, Option A - 'careful' is an adjective which can modify a noun, but not the verb.
Option C - 'cared' is verb form in its past tense, which cannot modify the verb.
Option D - 'carefulness' is noun form of the adjective 'serious' which cannot modify a verb.
Therefore, the suitable option to complete the given sentence is : Option B - 'carefully'.
The completed sentence, thus, becomes : She reads the text carefully before answering.