To solve the equation
\[ 3m + \frac{5}{4} - \frac{3m}{4} = -2, \]
let's first combine the terms involving \(m\). We can rewrite \(3m\) as \(\frac{12m}{4}\), allowing us to express \(3m\) and \(-\frac{3m}{4}\) with a common denominator:
\[ 3m = \frac{12m}{4}. \]
Now, substituting this back into the equation gives:
\[ \frac{12m}{4} - \frac{3m}{4} + \frac{5}{4} = -2. \]
Combining the \(m\) terms, we have:
\[ \frac{12m - 3m}{4} + \frac{5}{4} = -2. \]
This simplifies to:
\[ \frac{9m}{4} + \frac{5}{4} = -2. \]
Next, we can combine the fractions on the left:
\[ \frac{9m + 5}{4} = -2. \]
To eliminate the fraction, we can multiply both sides of the equation by \(4\):
\[ 9m + 5 = -8. \]
Next, we isolate \(9m\) by subtracting \(5\) from both sides:
\[ 9m = -8 - 5, \] \[ 9m = -13. \]
Now, we can solve for \(m\) by dividing both sides by \(9\):
\[ m = \frac{-13}{9}. \]
Thus, the solution is
\[ \boxed{\frac{-13}{9}}. \]