In a square area model, one part has area a^2, two rectangular parts have area ab each, and one part has area b^2. Which algebraic identity does this model represent?

Correct answer: (a+b)^2=a^2+2ab+b^2

The direct answer is A: (a+b)^2=a^2+2ab+b^2. The model contains one a^2 square, two rectangles of area ab, and one b^2 square. Adding their areas gives a^2+ab+ab+b^2=a^2+2ab+b^2. Because the outer figure is a square made by joining lengths a and b along each side, its side length is a+b and its area is (a+b)^2. Hence the two descriptions of the same total area are equal, giving option A. B has negative cross-regions and represents (a-b)^2, not the stated positive areas. C and D are equivalent forms of the difference-of-squares identity and require factors a+b and a-b; they do not match this four-part plus-area model. Be careful not to count only one ab rectangle: the two cross-regions together are 2ab. This is why the coefficient of the middle term is 2.