Explanation:
a "zero" is a value of x that leads to f(x) = 0.
since the function is already written as the multiplication of various terms of x (factorization), the zeroes can be easily identified.
remember, 0×factor = 0.
so, we need to find all values of x that make at least one of the factors 0. to be precise, all REAL values of x.
let's go through :
the first term is
(x + 2)²
that is actually two factors because of the square :
(x + 2)(x + 2)
both have the same zero solution : x = -2
and that's why x = -2 counts as 2 zeroes, even though they have the same value.
the next term is
(x² + 1)
x² + 1 = 0
x² = -1
x = ±sqrt(-1)
but sqrt(-1) is not a real number. it is the imaginary number i.
so, these 2 zeroes (±i) are not applying to or answer.
the next term is
(x - 16)
that is zero for x = 16
so, the real zeroes are -2, -2, 16.
or
-2, 16.
it depends on how your teacher wants you to document such multiple zeroes.