1. **State the problem:** Simplify the expression $$\frac{6 + 4(7 + 4)}{5^3 + 9}$$. 2. **Apply the order of operations (PEMDAS):** Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). 3. **Simplify inside the parentheses:** $$7 + 4 = 11$$ 4. **Multiply:** $$4 \times 11 = 44$$ 5. **Add in the numerator:** $$6 + 44 = 50$$ 6. **Calculate the denominator:** $$5^3 = 5 \times 5 \times 5 = 125$$ 7. **Add in the denominator:** $$125 + 9 = 134$$ 8. **Write the fraction:** $$\frac{50}{134}$$ 9. **Simplify the fraction by dividing numerator and denominator by their greatest common divisor (2):** $$\frac{\cancel{50}^{{\div 2}}}{\cancel{134}^{{\div 2}}} = \frac{25}{67}$$ **Final answer:** $$\frac{25}{67}$$