To find the [tex]$x$[/tex]-intercepts of the quadratic function [tex]\( g(x) = -2(x-4)(x+1) \)[/tex], we proceed as follows:
1. Identify the form of the quadratic function:
The function is given in factored form: [tex]\( g(x) = -2(x-4)(x+1) \)[/tex].
2. Set the function equal to zero to find the [tex]$x$[/tex]-intercepts:
The [tex]$x$[/tex]-intercepts are the values of [tex]\( x \)[/tex] that make [tex]\( g(x) \)[/tex] equal to zero. Thus, we set the equation:
[tex]\[
-2(x-4)(x+1) = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
For the product to be zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x - 4 = 0
\][/tex]
[tex]\[
x = 4
\][/tex]
and
[tex]\[
x + 1 = 0
\][/tex]
[tex]\[
x = -1
\][/tex]
4. Determine the [tex]$x$[/tex]-intercepts:
The solutions to the equations are [tex]\( x = 4 \)[/tex] and [tex]\( x = -1 \)[/tex]. These are the [tex]$x$[/tex]-coordinates where the graph of [tex]\( g(x) \)[/tex] intersects the [tex]$x$[/tex]-axis. Therefore, the [tex]$x$[/tex]-intercepts are:
[tex]\[
(4, 0) \quad \text{and} \quad (-1, 0)
\][/tex]
Thus, the correct answer is:
C. [tex]\((4,0)\)[/tex] and [tex]\((-1,0)\)[/tex]
