To determine the [tex]\(x\)[/tex]-intercepts of the quadratic function [tex]\(g(x) = -2(x-4)(x+1)\)[/tex], we need to solve for the values of [tex]\(x\)[/tex] that make [tex]\(g(x) = 0\)[/tex].
1. Set the function equal to zero:
[tex]\[
-2(x - 4)(x + 1) = 0
\][/tex]
2. Factor the equation:
[tex]\[
-2(x - 4)(x + 1) = 0
\][/tex]
Since the product of two factors is zero, one or both of the factors must be zero.
3. Set each factor equal to zero and solve for [tex]\(x\)[/tex]:
- For the factor [tex]\((x - 4) = 0\)[/tex]:
[tex]\[
x - 4 = 0 \implies x = 4
\][/tex]
- For the factor [tex]\((x + 1) = 0\)[/tex]:
[tex]\[
x + 1 = 0 \implies x = -1
\][/tex]
4. Determine the [tex]\(x\)[/tex]-intercepts:
The [tex]\(x\)[/tex]-intercepts occur at the points where the function crosses the [tex]\(x\)[/tex]-axis, which are the points [tex]\((4, 0)\)[/tex] and [tex]\((-1, 0)\)[/tex].
5. Select the correct option:
From the given options, the correct answer is:
B. [tex]\((4, 0)\)[/tex] and [tex]\((-1, 0)\)[/tex]
Thus, the [tex]\(x\)[/tex]-intercepts of the quadratic function [tex]\(g(x) = -2(x-4)(x+1)\)[/tex] are [tex]\((4, 0)\)[/tex] and [tex]\((-1, 0)\)[/tex].
