While making the visual model of \((a+b+c)^2\), which term appears twice?

Correct answer: \(ab\)

The direct answer is A: \(ab\) appears twice. To understand this visually, imagine a large square whose side is divided into lengths \(a\), \(b\), and \(c\) in both directions. The nine smaller regions include three squares, \(a^2\), \(b^2\), and \(c^2\), and six rectangles. The rectangle with sides \(a\) and \(b\) occurs in two positions: one has area \(ab\), and the other has area \(ba\). Since multiplication is commutative, \(ab=ba\), so together they contribute \(2ab\). Option A is correct. Option B is wrong because the \(a\)-by-\(a\) square occurs only once. Option C is wrong because the \(b\)-by-\(b\) square occurs only once. Option D is wrong because the \(c\)-by-\(c\) square also occurs only once. In fact, \((a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca\). Memory cue: same-by-same gives one square; different-by-different gives a pair of equal rectangles.