To find the [tex]$x$[/tex]-intercepts of the quadratic function [tex]\( f(x) = (x - 4)(x + 2) \)[/tex], we need to determine the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex].
### Step-by-Step Solution:
1. Set the function equal to zero:
[tex]\[
f(x) = (x - 4)(x + 2) = 0
\][/tex]
2. Determine the values of [tex]\( x \)[/tex] that make the product zero:
According to the zero-product property, if the product of two factors is zero, at least one of the factors must be zero. Therefore, we set each factor equal to zero separately and solve for [tex]\( x \)[/tex]:
[tex]\[
x - 4 = 0 \quad \text{or} \quad x + 2 = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
x - 4 = 0 \implies x = 4
\][/tex]
[tex]\[
x + 2 = 0 \implies x = -2
\][/tex]
4. Identify the [tex]$x$[/tex]-intercepts:
The [tex]$x$[/tex]-intercepts are the points at which [tex]\( f(x) = 0 \)[/tex]. These points are:
[tex]\[
(4, 0) \quad \text{and} \quad (-2, 0)
\][/tex]
### Conclusion:
Among the given options:
- [tex]$(-4, 0)$[/tex]
- [tex]$(-2, 0)$[/tex]
- [tex]$(0, 2)$[/tex]
- [tex]$(4, -2)$[/tex]
The points [tex]$(-2, 0)$[/tex] and [tex]$(4, 0)$[/tex] are the correct [tex]$x$[/tex]-intercepts of the quadratic function.
Therefore, the correct answer from the provided choices is:
[tex]\[ \boxed{(-2, 0)} \][/tex]
