To simplify the equation

\[
\frac{3}{x+3} - \frac{1}{x-2} = \frac{5}{2x+6}
\]

follow these steps:

1. **Rewrite \(2x + 6\)**:
\[
2x + 6 = 2(x + 3)
\]
So, we can rewrite the right-hand side:
\[
\frac{5}{2(x+3)}
\]

2. **Find a common denominator** for the left-hand side:
The common denominator for \((x + 3)(x - 2)\).

3. **Rewrite the left-hand side**:
\[
\frac{3(x - 2) - 1(x + 3)}{(x + 3)(x - 2)} = \frac{3x - 6 - x - 3}{(x + 3)(x - 2)} = \frac{2x - 9}{(x + 3)(x - 2)}
\]

4. **Now set the equation**:
\[
\frac{2x - 9}{(x + 3)(x - 2)} = \frac{5}{2(x + 3)}
\]

5. **Cross-multiply:**
\[
2(2x - 9) = 5(x - 2)
\]

Expanding both sides:
\[
4x - 18 = 5x - 10
\]

6. **Rearranging gives:**
\[
4x - 5x = -10 + 18
\]
\[
-x = 8 \implies x = -8
\]

Thus, the solution is:

\[
x = -8
\]