To solve the equation \((x + 6)(5^x - 1) = 0\), we set each factor in the equation to zero:

1. \(x + 6 = 0\)
2. \(5^x - 1 = 0\)

**Solving the first equation:**
\[
x + 6 = 0 \\
x = -6
\]

**Solving the second equation:**
\[
5^x - 1 = 0 \\
5^x = 1
\]
Since \(5^x = 1\) when \(x = 0\), we find:
\[
x = 0
\]

Now, we have the solutions from each factor:
1. \(x = -6\)
2. \(x = 0\)

Therefore, the complete solution to the equation \((x + 6)(5^x - 1) = 0\) is:
\[
\boxed{-6 \text{ and } 0}
\]