Solve the equation by completing the square.

To solve the equation \( z^2 + 10z + 6 = 0 \) by completing the square, follow these steps:

1. **Rearrange the equation** into a suitable form:

\[
z^2 + 10z = -6
\]

2. **Complete the square** for the left side. To do this, take half of the coefficient of \( z \) (which is 10), square it, and add it to both sides.

- Half of 10 is 5.
- Squaring 5 gives \( 25 \).

So, we add \( 25 \) to both sides:

\[
z^2 + 10z + 25 = -6 + 25
\]
\[
z^2 + 10z + 25 = 19
\]

3. **Factor the left side**:

\[
(z + 5)^2 = 19
\]

4. **Take the square root** of both sides:

\[
z + 5 = \pm \sqrt{19}
\]

5. **Solve for \( z \)**:

\[
z = -5 \pm \sqrt{19}
\]

Thus, the solutions to the equation \( z^2 + 10z + 6 = 0 \) are:

\[
z = -5 + \sqrt{19} \quad \text{and} \quad z = -5 - \sqrt{19}
\]

Among the provided options, the correct answer is:

**C.** \( -5 \pm \sqrt{19} \)