To solve the equation \((8 - x)^3 \cdot (3x + 6) = 0\), 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:
\[
(8 - x)^3 = 0
\]
Taking the cube root of both sides, we get:
\[
8 - x = 0
\]
Solving for \(x\), we find:
\[
x = 8
\]

2. Set the second factor to zero:
\[
3x + 6 = 0
\]
Solving for \(x\), we get:
\[
3x = -6
\]
\[
x = -2
\]

Therefore, the solutions to the equation \((8 - x)^3 \cdot (3x + 6) = 0\) are:
\[
x = 8 \quad \text{and} \quad x = -2
\]