Expanding Brackets and Conjugate Surds

Expand a single bracket of surds by distribution, expand two surd binomials with FOIL, square a surd binomial without dropping the cross term, and recognise the conjugate pattern $(\sqrt{x} + \sqrt{y})(\sqrt{x} - \sqrt{y}) = x - y$ — the difference-of-squares cancellation that always leaves a rational number.

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