Which expression is equivalent to (z + 4) (z - 4)?

To find the expression equivalent to \((z + 4)(z - 4)\), we can use the difference of squares formula, which states:

\[
(a + b)(a - b) = a^2 - b^2
\]

In this case, we can let \(a = z\) and \(b = 4\). Applying the formula:

\[
(z + 4)(z - 4) = z^2 - 4^2 = z^2 - 16
\]

Now let's evaluate the options provided:

1. **\(z^2 + 4z - 4z - 16\)**:
- Simplifying this gives \(z^2 - 16\), which matches our result.

2. **\(2z^2 - z^2 - 8\)**:
- Simplifying this gives \(z^2 - 8\), which does not match.

3. **\(z^2 - 8z - 16\)**:
- This does not match either, as it has an incorrect linear term.

4. **\(z(z) + 4z(z) - 4(4)\)**:
- Simplifying this gives \(z^2 + 4z^2 - 16 = 5z^2 - 16\), which also does not match.

Therefore, the correct expression that is equivalent to \((z + 4)(z - 4)\) is:

\[
z^2 - 16
\]

Out of the options provided, the first option **\(z^2 + 4z - 4z - 16\)** simplifies correctly to \(z^2 - 16\) and is equivalent to the expression we derived.