To solve the equation \((2x + 1)^3 = -\frac{1}{27}\), follow these steps:

1. Take the cube root of both sides of the equation to isolate \(2x + 1\):

\[
\sqrt[3]{(2x + 1)^3} = \sqrt[3]{-\frac{1}{27}}
\]

This simplifies to:

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

2. Solve for \(x\) by isolating \(x\):

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

Subtract 1 from both sides:

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

Convert 1 to a fraction with a common denominator:

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

Combine the fractions:

\[
2x = -\frac{4}{3}
\]

Divide both sides by 2:

\[
x = -\frac{4}{3} \div 2 = -\frac{4}{3} \times \frac{1}{2} = -\frac{4}{6} = -\frac{2}{3}
\]

So, the solution to the equation \((2x + 1)^3 = -\frac{1}{27}\) is:

\[
x = -\frac{2}{3}
\]