The correct answers are We had been preparing the vegetables for hours when the power suddenly went out. and The soup boiled over because the new cook had been ignoring the stove.

We use the past perfect continuous to talk about an action that was in progress up to a certain point in the past.

"Had been worked" is the wrong form (it should be "had been working"). "Had been dropping three plates" is incorrect because we use the past perfect simple (had dropped), not continuous, when we mention the specific quantity of completed actions.