A square side is divided into parts \(a\) and \(b\). The model has regions \(a^2\), \(b^2\), and two rectangles each of area \(ab\). Which identity does it represent?

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

The direct answer is A. The complete side of the square is formed by joining lengths \(a\) and \(b\), so it is \(a+b\). Its total area is therefore \((a+b)^2\). The model partitions that area into four parts: \(a^2\), \(ab\), another \(ab\), and \(b^2\). Adding them gives \(a^2+ab+ab+b^2=a^2+2ab+b^2\). Option B has \(-2ab\), which would correspond to subtracting mixed regions, not the two positive rectangles shown. Option C is the difference-of-squares identity and has a subtraction between the square areas. Option D is a cube identity and would require volume-like terms, not four regions of a square. Memory cue: a square split into a and b with two positive cross-rectangles gives plus, plus two-ab.