To solve the system of equations

\[
\begin{cases}
2x - y = -2 \\
x + 2y = 3m + 4
\end{cases}
\]

we can use substitution or elimination methods. Here, I'll provide a step-by-step solution using substitution:

1. **Solve the first equation for \(y\)**:

\[
2x - y = -2 \implies y = 2x + 2
\]

2. **Substitute \(y\) in the second equation**:

\[
x + 2(2x + 2) = 3m + 4
\]

Simplifying this:

\[
x + 4x + 4 = 3m + 4 \\
5x + 4 = 3m + 4
\]

3. **Isolate \(x\)**:

\[
5x = 3m + 4 - 4 \\
5x = 3m \\
x = \frac{3m}{5}
\]

4. **Now substitute \(x\) back into the equation for \(y\)**:

\[
y = 2\left(\frac{3m}{5}\right) + 2 \\
y = \frac{6m}{5} + 2
\]

To have a common denominator:

\[
y = \frac{6m}{5} + \frac{10}{5} = \frac{6m + 10}{5}
\]

Thus, the solution to the system of equations is:

\[
\begin{cases}
x = \frac{3m}{5} \\
y = \frac{6m + 10}{5}
\end{cases}
\]