Answer
$$(z^2 + 4)(z + 2)(z - 2)$$
Work Step by Step
To factor the difference of two squares, we use the formula:
$$A^2 - B^2 = (A + B)(A - B)$$
For this binomial, we take the square root of the leading term $(z^4)$ to get $A = z^2$ and the square root of the second term $(16)$ to get $B = 4$.
So we now plug these two values into the formula for factoring the binomial:
$$(z^2 + 4)(z^2 - 4)$$
The second binomial is also a difference of two squares, so we can use the same formula to further factor this problem:
To factor the difference of two squares, we use the formula:
$$A^2 - B^2 = (A + B)(A - B)$$
For this binomial, we take the square root of the leading term $(z^2)$ to get $A = z$ and the square root of the second term $(4)$ to get $B = 2$.
We plug these values into the equation to factor the difference of two squares to get:
$$(z + 2)(z - 2)$$
We now add what we just factored to replace the binomial $(z^2 - 4)$ to get the final factorization of the original binomial:
$$(z^2 + 4)(z + 2)(z - 2)$$
