To solve the equation \(\frac{x}{6} + \frac{x}{12} + \frac{x}{20} + \frac{x}{30} = -\frac{5}{9}\), we first need to find a common denominator for the fractions on the left-hand side. The denominators are 6, 12, 20, and 30.

The least common multiple (LCM) of these numbers is 60. We will convert each fraction to have this common denominator:

\[
\frac{x}{6} = \frac{x \cdot 10}{60} = \frac{10x}{60}
\]

\[
\frac{x}{12} = \frac{x \cdot 5}{60} = \frac{5x}{60}
\]

\[
\frac{x}{20} = \frac{x \cdot 3}{60} = \frac{3x}{60}
\]

\[
\frac{x}{30} = \frac{x \cdot 2}{60} = \frac{2x}{60}
\]

Now, we can rewrite the equation with these fractions:

\[
\frac{10x}{60} + \frac{5x}{60} + \frac{3x}{60} + \frac{2x}{60} = -\frac{5}{9}
\]

Combine the fractions on the left-hand side:

\[
\frac{10x + 5x + 3x + 2x}{60} = \frac{20x}{60}
\]

Simplify the fraction:

\[
\frac{20x}{60} = \frac{x}{3}
\]

So the equation now is:

\[
\frac{x}{3} = -\frac{5}{9}
\]

To solve for \(x\), multiply both sides by 3:

\[
x = -\frac{5}{9} \cdot 3 = -\frac{15}{9} = -\frac{5}{3}
\]

Thus, the solution to the equation is:

\[
x = -\frac{5}{3}
\]