In direct variation, the relationship between two variables can be expressed in the form \( y = kx \), where \( k \) is the constant of variation.
We start with the equation given:
\[ 3x + 5y = 0 \]
To express \( y \) in terms of \( x \), we can rearrange the equation:
\[ 5y = -3x \]
Now, divide both sides by 5:
\[ y = -\frac{3}{5}x \]
Here, we can see that this equation is in the form \( y = kx \), where \( k = -\frac{3}{5} \).
Thus, the constant of variation \( k \) is
\[ \boxed{-\frac{3}{5}}. \]