1. **State the problem:** Solve the equation $$2\sqrt{7} + (x-2)(\sqrt{7} + x) = 0$$ for $x$. 2. **Expand the product:** Use the distributive property to expand $(x-2)(\sqrt{7} + x)$: $$ (x-2)(\sqrt{7} + x) = x \cdot \sqrt{7} + x \cdot x - 2 \cdot \sqrt{7} - 2 \cdot x = x\sqrt{7} + x^2 - 2\sqrt{7} - 2x $$ 3. **Rewrite the equation:** Substitute the expansion back into the original equation: $$ 2\sqrt{7} + x\sqrt{7} + x^2 - 2\sqrt{7} - 2x = 0 $$ 4. **Simplify terms:** Combine like terms: $$ 2\sqrt{7} - 2\sqrt{7} + x\sqrt{7} + x^2 - 2x = 0 $$ $$ x\sqrt{7} + x^2 - 2x = 0 $$ 5. **Rewrite the equation:** $$ x^2 + x\sqrt{7} - 2x = 0 $$ 6. **Factor out $x$:** $$ x(x + \sqrt{7} - 2) = 0 $$ 7. **Set each factor equal to zero:** - $x = 0$ - $x + \sqrt{7} - 2 = 0$ 8. **Solve the second equation:** $$ x = 2 - \sqrt{7} $$ **Final solutions:** $$ x = 0 \quad \text{or} \quad x = 2 - \sqrt{7} $$