A large square has side p + q, and inside it is a square of side p − q. What is the area of the border strip between them?

Correct answer: 4pq

The direct answer is B: 4pq. The border is the area of the outer square minus the area of the inner square. The outer area is (p+q)² and the inner area is (p−q)², so border area = (p+q)²−(p−q)². Apply a²−b²=(a−b)(a+b): [(p+q)−(p−q)] [(p+q)+(p−q)] = (2q)(2p)=4pq. Option A is only half of the complete result. Option C, p²−q², is the product (p−q)(p+q), which is related to the identity but is not the difference between these two square areas. Option D comes from adding rather than subtracting the square areas and is therefore wrong. A safe visual check is that the border must equal outer area minus inner area, not their sum.