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To solve the equation \(3m + \frac{5}{4} - \frac{3m}{4} = -2\), we can start by simplifying and isolating the variable \(m\).
First, let's rewrite the equation clearly:
\[
3m + \frac{5}{4} - \frac{3m}{4} = -2
\]
Next, we can express \(3m\) with a common denominator of 4:
\[
3m = \frac{12m}{4}
\]
So the equation becomes:
\[
\frac{12m}{4} - \frac{3m}{4} + \frac{5}{4} = -2
\]
Combining the fractions on the left gives:
\[
\frac{12m - 3m + 5}{4} = -2
\]
\[
\frac{9m + 5}{4} = -2
\]
Now, multiply both sides by 4 to eliminate the fraction:
\[
9m + 5 = -8
\]
Next, subtract 5 from both sides:
\[
9m = -8 - 5
\]
\[
9m = -13
\]
Now, divide both sides by 9:
\[
m = \frac{-13}{9}
\]
The value for \(m\) that makes the equation true is:
\[
\boxed{\frac{-13}{9}}
\]