1. **State the problem:** We need to find three fifths of the sum of four times a certain number and 9. 2. **Define the variable:** Let the certain number be $x$. 3. **Write the expression for the sum:** Four times the number plus 9 is $4x + 9$. 4. **Multiply by three fifths:** Three fifths of this sum is $\frac{3}{5}(4x + 9)$. 5. **Simplify the expression:** $$\frac{3}{5}(4x + 9) = \frac{3}{5} \times 4x + \frac{3}{5} \times 9 = \frac{12}{5}x + \frac{27}{5}$$ 6. **Final answer:** Three fifths the sum of four times a certain number and 9 is $$\frac{12}{5}x + \frac{27}{5}$$.