To determine the [tex]\(x\)[/tex]-intercepts of the quadratic function [tex]\(g(x) = -2(x-4)(x+1)\)[/tex], we need to find the points where the function crosses the [tex]\(x\)[/tex]-axis. This occurs when [tex]\(g(x) = 0\)[/tex].
Setting the function equal to zero, we have:
[tex]\[
-2(x-4)(x+1) = 0
\][/tex]
Next, we solve for the values of [tex]\(x\)[/tex] that make this equation true. We can use the Zero Product Property, which states that if a product of factors equals zero, then at least one of the factors must be zero.
We have two factors in our equation: [tex]\((x-4)\)[/tex] and [tex]\((x+1)\)[/tex]. We set each factor equal to zero and solve for [tex]\(x\)[/tex].
1. Set [tex]\(x-4=0\)[/tex]:
[tex]\[
x - 4 = 0 \implies x = 4
\][/tex]
2. Set [tex]\(x+1=0\)[/tex]:
[tex]\[
x + 1 = 0 \implies x = -1
\][/tex]
Thus, the [tex]\(x\)[/tex]-intercepts of the function are the points [tex]\((4, 0)\)[/tex] and [tex]\((-1, 0)\)[/tex].
Therefore, the correct answer is:
C. [tex]\((4,0)\)[/tex] and [tex]\((-1,0)\)[/tex]
