1. necesitan escribir un libro.
a. incorrect
b. correct

2. ellos (poder) hablar en alemán.
a. puedo
b. poder
c. pueden
d. podemos

3. ¿qué necesitas hacer hoy?
a. necesito estudiar.
b. necesitar estudiar.
c. necesito estudio.
d. necesito estudiamos.

4. ¿tú qué acabas de hacer?
a. acabar de ir a la escuea.
b. acabo de ir a la escuela.
c. acaba de ir a la escuela.

This sentence is correct and stands for the simple present. The simple present tense is used to talk about habitual actions or routines. This tense is also used to describe things that happen now or in the near future. So necesitan is the conjugation of the verb necesitar for the third person plural in the simple present. In Spanish, it's common not to include the subject pronoun in a sentence. Therefore, from the conjugation we know that the subject pronoun here is ellos or ellas (they in English).

Here the subject pronoun has been included and is ellos (third person plural). Therefore, we must choose the conjugation of the verb poder that matches this subject pronoun. So puedo matches the first person singular (simple present), poder is an infinitive verb and podemos matches the first person plural (simple present). The only option that matches the third person singular (simple present) is pueden. Finally:

Ellos pueden hablar en alemán

Here we use the verb necesitar. From the question, the verb necesitar is conjugated matching the second person singular (tú) in the simple present. So the reply to this question must match the first person singular in the simple present, which is necesito. Then, the verb after this word needs to be in infinitive form (ar, er or ir ending). So the only option that meets this requirement is A. Finally:

Necesito estudiar

From the question, acabas is the conjugation of the verb acabar that matches the second person singular in the simple present. So the reply to this question must match the first person singular. Therefore, acabo matches this person in the simple present. Accordingly, option B) is the correct.