There's no such thing as "negative area" - it's just saying that this area lies below the x-axis.
So yes, if you have two curves below the x-axis, the area bounded between them also lies below the x-axis, and thus should have negative sign to denote this.
But if you're asked for the AREA, you can NOT write it with a negative sign, because there's no such thing as negative area. Remember, Area is "How many unit squares lie in this region". Can you have a negative number of unit squares? Of course not.
If you're asked for Area, you can not write a negative sign unless it specifically asks you to denote whether it lies above or below the x-axis.