To find the correct factored form of the expression [tex]\( 4(x^2 - 2x) - 2(x^2 - 3) \)[/tex], let's break down the problem step by step.
1. Expand the given expression:
[tex]\[
4(x^2 - 2x) - 2(x^2 - 3)
\][/tex]
First, distribute the constants inside the parentheses:
[tex]\[
4 \cdot x^2 - 4 \cdot 2x - 2 \cdot x^2 - 2 \cdot (-3)
\][/tex]
This simplifies to:
[tex]\[
4x^2 - 8x - 2x^2 + 6
\][/tex]
2. Combine like terms:
[tex]\[
(4x^2 - 2x^2) + (-8x) + 6
\][/tex]
Simplify the terms within each group:
[tex]\[
2x^2 - 8x + 6
\][/tex]
3. Factor the simplified expression [tex]\(2x^2 - 8x + 6\)[/tex]:
We want to factor this quadratic expression. To do this, look for two numbers that multiply to [tex]\(2 \times 6 = 12\)[/tex] and add up to [tex]\(-8\)[/tex].
The numbers [tex]\(-6\)[/tex] and [tex]\(-2\)[/tex] satisfy these conditions:
[tex]\[
2x^2 - 8x + 6 = 2x^2 - 6x - 2x + 6
\][/tex]
Factor by grouping:
[tex]\[
2x(x - 3) - 2(x - 3)
\][/tex]
Factor out the common factor [tex]\((x - 3)\)[/tex]:
[tex]\[
(2x - 2)(x - 3)
\][/tex]
Further factor out the 2 from [tex]\(2x - 2\)[/tex]:
[tex]\[
2(x - 1)(x - 3)
\][/tex]
Hence, the factored form of the expression [tex]\(4(x^2 - 2x) - 2(x^2 - 3)\)[/tex] is [tex]\(2(x - 1)(x - 3)\)[/tex].
The correct answer is:
[tex]\[ \boxed{2(x-1)(x-3)} \][/tex]
So, the correct choice is:
[tex]\[ \boxed{\text{C. } 2(x-1)(x-3)} \][/tex]
