Divide x⁶ - y⁶ by the product of x² + xy + y² and x - y.

The product of (x² + xy + y²) and (x - y)

= (x² + xy + y²) $\times$ (x - y)

= x $\times$ (x² + xy + y²) - y $\times$ (x² + xy + y²)

= x⁽¹⁺²⁾ + x⁽¹⁺¹⁾y + xy² - x²y - xy⁽¹⁺¹⁾ - y⁽¹⁺²⁾

= x³ + x²y + xy² - x²y - xy² - y³

= x³ + (x²y - x²y) + (xy² - xy²) - y³

= x³ - y³

Now, (x⁶ - y⁶) ÷ ( x³ - y³)

$\begin{array}{l} \phantom{x^3 - y^3)}{x^3 + y^3} \ x^3 - y^3\overline{\smash{\big)}x^6 - y^6 \phantom{+ x^3y^3}} \ \phantom{x^3 - y^3}\underline{\underset{-}{}x^6 \phantom{- y^6} \underset{+}{-}x^3y^3} \ \phantom{{x^3 - y^3}6x^2 +}x^3y^3 - y^6 \ \phantom{{x^3 - y^3}6x^21}\underline{\underset{-}{+}x^3y^3 \underset{+}{-} y^6} \ \phantom{{x^3 - y^3}6x^2 + 9x -} \times \end{array}$

Hence, the answer is (x³ + y³)