To solve the equation \((4x + 2)(x^2 + 1) = 0\), we can set each factor equal to zero.

1. **First factor:**
\[
4x + 2 = 0
\]
Solving for \(x\):
\[
4x = -2 \quad \Rightarrow \quad x = -\frac{2}{4} = -\frac{1}{2}
\]

2. **Second factor:**
\[
x^2 + 1 = 0
\]
Solving for \(x\):
\[
x^2 = -1
\]
This produces complex solutions:
\[
x = \pm i
\]

**Summary of solutions:**
- The real solution is \(x = -\frac{1}{2}\).
- The complex solutions are \(x = i\) and \(x = -i\).