To solve the system of equations:

\[
\begin{cases}
3x + 4y = -37 \\
2x + y = 17
\end{cases}
\]

we can use the substitution or elimination method. Here, we will use the elimination method.

First, let's multiply the second equation by 4 to align the coefficients of \( y \):

\[
4(2x + y) = 4(17)
\]

This gives us:

\[
8x + 4y = 68
\]

Now, we have the system:

\[
\begin{cases}
3x + 4y = -37 \\
8x + 4y = 68
\end{cases}
\]

Next, subtract the first equation from the second equation to eliminate \( y \):

\[
(8x + 4y) - (3x + 4y) = 68 - (-37)
\]

Simplifying this, we get:

\[
8x + 4y - 3x - 4y = 68 + 37
\]

\[
5x = 105
\]

\[
x = \frac{105}{5}
\]

\[
x = 21
\]

Now, substitute \( x = 21 \) back into the second original equation to find \( y \):

\[
2(21) + y = 17
\]

\[
42 + y = 17
\]

\[
y = 17 - 42
\]

\[
y = -25
\]

So, the solution to the system of equations is:

\[
x = 21, \quad y = -25
\]