How much longer...... this book?

The correct phrase to complete the sentence is "will you be needing."

So, the complete sentence is:

"How much longer will you be needing this book?"

This phrase is used to ask about the duration for which the book will be needed in the future.

Here's why the other options are incorrect:

• "you are needing": This is not the correct form for future context; "are needing" is not commonly used in English.
• "will you have needed": This suggests a future perfect tense, which is not appropriate for asking about duration in the present or near future.
• "have you needed": This is in the present perfect tense, which is used to talk about something that started in the past and continues to the present, but does not fit the context of asking about future duration.