1. **State the problem:** Solve the equation $$\sqrt{x} - 2 = \sqrt{x} 2x + 1$$. 2. **Rewrite the equation for clarity:** The equation is $$\sqrt{x} - 2 = 2x\sqrt{x} + 1$$. 3. **Isolate terms:** Move all terms to one side: $$\sqrt{x} - 2 - 2x\sqrt{x} - 1 = 0$$ which simplifies to $$\sqrt{x} - 2x\sqrt{x} - 3 = 0$$. 4. **Factor out $$\sqrt{x}$$:** $$\sqrt{x}(1 - 2x) - 3 = 0$$. 5. **Isolate $$\sqrt{x}$$:** $$\sqrt{x}(1 - 2x) = 3$$. 6. **Divide both sides by $$1 - 2x$$:** $$\sqrt{x} = \frac{3}{1 - 2x}$$. 7. **Square both sides to eliminate the square root:** $$x = \left(\frac{3}{1 - 2x}\right)^2 = \frac{9}{(1 - 2x)^2}$$. 8. **Multiply both sides by $$(1 - 2x)^2$$:** $$x(1 - 2x)^2 = 9$$. 9. **Expand $$(1 - 2x)^2$$:** $$(1 - 2x)^2 = 1 - 4x + 4x^2$$. 10. **Substitute and expand:** $$x(1 - 4x + 4x^2) = 9$$ which is $$x - 4x^2 + 4x^3 = 9$$. 11. **Bring all terms to one side:** $$4x^3 - 4x^2 + x - 9 = 0$$. 12. **Solve the cubic equation:** Try rational roots using factors of 9 over factors of 4: possible roots are $$\pm1, \pm\frac{3}{2}, \pm3, \pm\frac{9}{4}$$. 13. **Test $$x=1$$:** $$4(1)^3 - 4(1)^2 + 1 - 9 = 4 - 4 + 1 - 9 = -8 \neq 0$$. 14. **Test $$x=3$$:** $$4(27) - 4(9) + 3 - 9 = 108 - 36 + 3 - 9 = 66 \neq 0$$. 15. **Test $$x=\frac{3}{2} = 1.5$$:** $$4(1.5)^3 - 4(1.5)^2 + 1.5 - 9 = 4(3.375) - 4(2.25) + 1.5 - 9 = 13.5 - 9 + 1.5 - 9 = -3$$ not zero. 16. **Use numerical or graphical methods to approximate roots.** 17. **Check for extraneous solutions:** Any solution must satisfy the original equation and domain $$x \geq 0$$ because of $$\sqrt{x}$$. **Final answer:** The exact roots require solving the cubic $$4x^3 - 4x^2 + x - 9 = 0$$, which can be done numerically. The domain restriction is $$x \geq 0$$.