1. **State the problem:** We need to write an algebraic expression for "Three-fifth the sum of 4 times a certain number and 9". 2. **Identify the variable:** Let the certain number be $x$. 3. **Write the sum inside the parentheses:** The sum of 4 times the number and 9 is $4x + 9$. 4. **Multiply the sum by three-fifths:** Multiply the entire sum by $\frac{3}{5}$. 5. **Final expression:** $$\frac{3}{5}(4x + 9)$$ This expression represents three-fifths of the sum of four times a certain number and 9.