Two matching strips almost form a bigger square. One small corner completes it.
Expanding (x+b)(x+b) gives x²+bx+bx+b². Thus x²+2bx=(x+b)²−b². For x²+cx, the required strip width is c/2 and the added term is (c/2)². The geometric picture uses positive lengths; the identity is algebraic.
Completing a square cannot change an equation on only one side.