1. **State the problem:** Simplify the expression $$\frac{5 + 9(1 + 2)^2}{1 + 9}$$. 2. **Recall the order of operations:** Parentheses first, then exponents, multiplication/division, and addition/subtraction. 3. **Calculate inside the parentheses:** $$1 + 2 = 3$$. 4. **Apply the exponent:** $$(3)^2 = 9$$. 5. **Multiply:** $$9 \times 9 = 81$$. 6. **Add in the numerator:** $$5 + 81 = 86$$. 7. **Calculate the denominator:** $$1 + 9 = 10$$. 8. **Form the fraction:** $$\frac{86}{10}$$. 9. **Simplify the fraction by dividing numerator and denominator by 2:** $$\frac{\cancel{86}^{43}}{\cancel{10}^{5}} = \frac{43}{5}$$. 10. **Final answer:** $$\frac{43}{5}$$ or 8.6.