The correct answer is E. bought/has been bragging.
The sentence is in the present perfect continuous tense, which is used to talk about an action that started in the past and continues up to the present. In this case, the action is “bragging,” and it started when the brother bought the new BMW. The sentence also tells us that the brother has been bragging about the car for a long time, so we can use the present perfect continuous tense to emphasize that.
The other options are incorrect because they do not use the present perfect continuous tense. Option A, “was buying/bragged,” is in the past progressive tense, which is used to talk about an action that was happening at a specific time in the past. Option B, “is buying/is bragging,” is in the present progressive tense,
which is used to talk about an action that is happening right now. Option C, “has bought/brags,” is in the simple past tense, which is used to talk about an action that happened at a specific time in the past. Option D, “buys/has bragged,” is in the simple present tense, which is used to talk about an action that happens regularly or repeatedly.In conclusion, the correct answer is E. bought/has been bragging.