To solve the equation \((x^2 - 16)(x + 3) = 0\), we need to find the values of \(x\) that make the equation true. This can be done by setting each factor equal to zero and solving for \(x\).

1. **First Factor: \(x^2 - 16 = 0\)**

\[
x^2 - 16 = 0
\]

This is a difference of squares, which can be factored further:

\[
(x - 4)(x + 4) = 0
\]

Setting each factor equal to zero gives us:

\[
x - 4 = 0 \quad \text{or} \quad x + 4 = 0
\]

Solving these equations:

\[
x = 4 \quad \text{or} \quad x = -4
\]

2. **Second Factor: \(x + 3 = 0\)**

\[
x + 3 = 0
\]

Solving for \(x\):

\[
x = -3
\]

So, the solutions to the equation \((x^2 - 16)(x + 3) = 0\) are:

\[
x = 4, \quad x = -4, \quad \text{and} \quad x = -3
\]