(a + b)² = a² + 2ab + b² is first a fact about tiling.
A square whose side is the sum of two segments splits cleanly into a square of the first, a square of the second, and two rectangles that share those lengths.
The identical dissection with a minus sign produces the twin identity (a − b)² = a² − 2ab + b².
Both rest on the same principle that later rearranges the region a² − b² into the single rectangle (a + b)(a − b).
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