1. **State the problem:** Simplify the expression $$\frac{0.1 + (1 - 0.1)}{1 + 2\sqrt{7}} - \frac{1}{-1 + 2\sqrt{7}} + \frac{2}{-1 + 1.9} + \frac{3}{?}$$ (Note: The last denominator is unclear, so we will simplify the first two terms clearly given the input.) 2. **Rewrite the expression clearly:** $$\frac{0.1 + 0.9}{1 + 2\sqrt{7}} - \frac{1}{-1 + 2\sqrt{7}} + \frac{2}{-1 + 1.9}$$ 3. **Simplify the numerators:** $$\frac{1}{1 + 2\sqrt{7}} - \frac{1}{-1 + 2\sqrt{7}} + \frac{2}{0.9}$$ 4. **Rationalize the denominators of the first two fractions:** For $$\frac{1}{1 + 2\sqrt{7}}$$ multiply numerator and denominator by $$1 - 2\sqrt{7}$$: $$\frac{1 \cdot (1 - 2\sqrt{7})}{(1 + 2\sqrt{7})(1 - 2\sqrt{7})} = \frac{1 - 2\sqrt{7}}{1 - (2\sqrt{7})^2} = \frac{1 - 2\sqrt{7}}{1 - 4 \cdot 7} = \frac{1 - 2\sqrt{7}}{1 - 28} = \frac{1 - 2\sqrt{7}}{-27}$$ For $$\frac{1}{-1 + 2\sqrt{7}}$$ multiply numerator and denominator by $$-1 - 2\sqrt{7}$$: $$\frac{1 \cdot (-1 - 2\sqrt{7})}{(-1 + 2\sqrt{7})(-1 - 2\sqrt{7})} = \frac{-1 - 2\sqrt{7}}{(-1)^2 - (2\sqrt{7})^2} = \frac{-1 - 2\sqrt{7}}{1 - 28} = \frac{-1 - 2\sqrt{7}}{-27}$$ 5. **Rewrite the expression:** $$\frac{1 - 2\sqrt{7}}{-27} - \frac{-1 - 2\sqrt{7}}{-27} + \frac{2}{0.9}$$ 6. **Combine the first two fractions since they have the same denominator:** $$\frac{1 - 2\sqrt{7} - (-1 - 2\sqrt{7})}{-27} + \frac{2}{0.9} = \frac{1 - 2\sqrt{7} + 1 + 2\sqrt{7}}{-27} + \frac{2}{0.9}$$ Simplify numerator: $$1 - 2\sqrt{7} + 1 + 2\sqrt{7} = 2$$ So the fraction is: $$\frac{2}{-27} = -\frac{2}{27}$$ 7. **Simplify the last term:** $$\frac{2}{0.9} = \frac{2}{\frac{9}{10}} = 2 \times \frac{10}{9} = \frac{20}{9}$$ 8. **Final expression:** $$-\frac{2}{27} + \frac{20}{9}$$ Convert to common denominator 27: $$-\frac{2}{27} + \frac{20 \times 3}{27} = -\frac{2}{27} + \frac{60}{27} = \frac{58}{27}$$ **Answer:** $$\boxed{\frac{58}{27}}$$