If the outer square has side x + 5 and the inner square has side x − 1, what is the border area?

Correct answer: 12x + 24

The direct answer is A: the border area is 12x + 24. The border is the part inside the outer square but outside the inner square, so subtract the two areas: (x + 5)² − (x − 1)². Expand carefully: (x + 5)² = x² + 10x + 25, and (x − 1)² = x² − 2x + 1. Therefore, the border area is x² + 10x + 25 − x² + 2x − 1 = 12x + 24. A is correct. B and C have incorrect coefficients of x; they result from incomplete or mistaken subtraction. D is only the area of the outer square, so it includes the inner square and is not the border alone. The same result follows from the difference-of-squares identity: [(x + 5) + (x − 1)] [(x + 5) − (x − 1)] = (2x + 4)(6) = 12x + 24. Memory cue: border area means outside area minus inside area.