The correct option is
A die
In order to fill in the blank with the most suitable form of the infinitive, we need to understand the meaning and usage of 'would rather'. 'Would rather' means the same as 'rather than', which means 'instead of' or preferring someone or something over the other. If 'would rather' is used in a passive (voice) sentence, we use the word 'to' along with the verb. In an active sentence, we do not use 'to' with the verb. The verbs without the marker 'to', are called bare infinitives.As the given sentence is in the active voice, we will use the bare infinitive here.
Option A - 'Dead' is the past form of the verb 'die'. As we need the bare infinitive (original form of the verb), we can't use 'dead' here. Hence, option A is incorrect.
Option C -'Dying' is the participle form of 'die'. As we need the verb in its original form, we can't use 'dying' here. Hence, option C is incorrect.
Option D - 'To die' can't be used here as the verb in its original form with the marker 'to'. As the sentence is in the active voice, we can't use the infinitive with 'to' here. Hence, option D is incorrect.
Option B - 'Die' is a verb in its original form and is a bare infinitive. Hence, option B is the correct answer.
'She would rather die than suffer so.'