1. **State the problem:** Simplify the expression $$\frac{6}{\sqrt{7} + 1}$$. 2. **Formula and rule:** To simplify a fraction with a surd in the denominator, multiply numerator and denominator by the conjugate of the denominator. The conjugate of $$\sqrt{7} + 1$$ is $$\sqrt{7} - 1$$. 3. **Multiply numerator and denominator by the conjugate:** $$\frac{6}{\sqrt{7} + 1} \times \frac{\sqrt{7} - 1}{\sqrt{7} - 1} = \frac{6(\sqrt{7} - 1)}{(\sqrt{7} + 1)(\sqrt{7} - 1)}$$ 4. **Simplify the denominator using difference of squares:** $$ (\sqrt{7})^2 - 1^2 = 7 - 1 = 6 $$ 5. **Substitute back:** $$ \frac{6(\sqrt{7} - 1)}{6} $$ 6. **Cancel the common factor 6:** $$ \frac{\cancel{6}(\sqrt{7} - 1)}{\cancel{6}} = \sqrt{7} - 1 $$ 7. **Final answer:** $$ \boxed{\sqrt{7} - 1} $$