To determine how many solutions the equation \( \frac{2}{3}(6m+9) - 6 = 4m \) has, we will first simplify and solve it step-by-step.
Starting with the equation:
\[ \frac{2}{3}(6m + 9) - 6 = 4m \]
Step 1: Distribute \( \frac{2}{3} \) to the terms inside the parentheses.
\[ \frac{2}{3} \cdot 6m + \frac{2}{3} \cdot 9 - 6 = 4m \]
This simplifies to:
\[ 4m + 6 - 6 = 4m \]
Step 2: Simplify the left side:
\[ 4m = 4m \]
Now we see that both sides of the equation are identical, meaning that the equation is true for all values of \( m \).
Conclusion: Since this is a true statement for all \( m \), the equation has an infinite number of solutions.
So, the answer is: infinitely many solutions.