1. **Stating the problem:** Solve the quadratic equation $$x^2 = 2(\sqrt{11}x + x - 2\sqrt{11})$$. 2. **Expand the right side:** Use distributive property: $$x^2 = 2\sqrt{11}x + 2x - 4\sqrt{11}$$ 3. **Bring all terms to one side to set equation to zero:** $$x^2 - 2\sqrt{11}x - 2x + 4\sqrt{11} = 0$$ 4. **Group like terms:** $$x^2 - (2\sqrt{11} + 2)x + 4\sqrt{11} = 0$$ 5. **Use quadratic formula:** For equation $$ax^2 + bx + c = 0$$, solutions are $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $$a=1$$, $$b=-(2\sqrt{11} + 2)$$, $$c=4\sqrt{11}$$. 6. **Calculate discriminant:** $$\Delta = b^2 - 4ac = (-(2\sqrt{11} + 2))^2 - 4 \cdot 1 \cdot 4\sqrt{11}$$ $$= (2\sqrt{11} + 2)^2 - 16\sqrt{11}$$ $$= (2\sqrt{11})^2 + 2 \cdot 2\sqrt{11} \cdot 2 + 2^2 - 16\sqrt{11}$$ $$= 4 \cdot 11 + 8\sqrt{11} + 4 - 16\sqrt{11}$$ $$= 44 + 4 + 8\sqrt{11} - 16\sqrt{11}$$ $$= 48 - 8\sqrt{11}$$ 7. **Calculate roots:** $$x = \frac{2\sqrt{11} + 2 \pm \sqrt{48 - 8\sqrt{11}}}{2}$$ 8. **Simplify by factoring 2 in numerator:** $$x = \frac{\cancel{2}(\sqrt{11} + 1) \pm \sqrt{48 - 8\sqrt{11}}}{\cancel{2} \cdot 2} = \frac{\sqrt{11} + 1 \pm \frac{\sqrt{48 - 8\sqrt{11}}}{2}}{1}$$ 9. **Final solutions:** $$x = \sqrt{11} + 1 \pm \frac{\sqrt{48 - 8\sqrt{11}}}{2}$$ This gives two real roots for the quadratic equation.