To find the equivalent expression to \((z + 4)(z - 4)\) using the Distributive Property, we can apply the formula for the difference of squares:
\[ (a + b)(a - b) = a^2 - b^2 \]
In this case, \(a = z\) and \(b = 4\). Thus, we can calculate:
\[ (z + 4)(z - 4) = z^2 - 4^2 = z^2 - 16 \]
Now, let's examine the options provided:
- \(z(z) + 4z(z) - 4(4)\) – This expression does not simplify correctly and is not equivalent to our Expression.
- \(z^2 - 8z - 16\) – This expression introduces an incorrect middle term (\(-8z\)), which is not part of the original expansion.
- \(z^2 + 4z - 4z - 16\) – This simplifies to \(z^2 - 16\) because \(4z - 4z = 0\), matching our result.
- \(2z^2 - z^2 - 8\) – This expression can be simplified to \(z^2 - 8\), which is not correct.
The correct equivalent expression to \((z + 4)(z - 4)\) is:
\[ \boxed{z^2 - 16} \]
This matches the simplified version we obtained using the difference of squares. The closest fitting option from the choices is option 3, \(z^2 + 4z - 4z - 16\), which simplifies properly to our answer.