To simplify the expression \((z+4)(z-4)\), we can use the difference of squares formula, which states:
\[ (a+b)(a-b) = a^2 - b^2 \]
In this case, \(a = z\) and \(b = 4\). Therefore, we can write:
\[ (z+4)(z-4) = z^2 - 4^2 = z^2 - 16 \]
Now let's review the answer choices to find an equivalent expression:
A. \(z^2 + 4z - 4z - 16\)
Combining like terms:
\[ z^2 + 4z - 4z - 16 = z^2 - 16 \]
B. \(z(z) + 4z(z) - 4(4)\)
This expands to:
\[ z^2 + 4z^2 - 16 = 5z^2 - 16 \quad \text{(not equivalent)} \]
C. \(z^2 - 8z - 16\)
This is clearly not equivalent to \(z^2 - 16\) due to the \(-8z\) term.
D. \(2z^2 - z^2 - 8\)
Combining like terms gives:
\[ 2z^2 - z^2 - 8 = z^2 - 8 \quad \text{(not equivalent)} \]
The only expression that is equivalent to \((z + 4)(z - 4)\) is:
A. \(z^2 - 16\).