To solve the equation \((3x - 3)^3 \cdot (x + 4) = 0\), we need to find the values of \(x\) that make the equation true. This equation is a product of two factors equal to zero, so we can set each factor to zero and solve for \(x\).

1. Set the first factor to zero:
\[
(3x - 3)^3 = 0
\]
To solve this, we take the cube root of both sides:
\[
3x - 3 = 0
\]
Solving for \(x\):
\[
3x = 3
\]
\[
x = 1
\]

2. Set the second factor to zero:
\[
x + 4 = 0
\]
Solving for \(x\):
\[
x = -4
\]

Therefore, the solutions to the equation \((3x - 3)^3 \cdot (x + 4) = 0\) are:
\[
x = 1 \quad \text{and} \quad x = -4
\]