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a² + b² - 6ab = (3 +√6)² + (3 - √6)² - 6(3 + √6)(3 -√6)
= 9 + 6√6 + 6 + 9 - 6√6 + 6 - 6(9 - 6)
= 9 + 6 + 9 + 6 - 18
= 12