To find the factored form of the given expression, we start with the expression:
[tex]\[
4(x^2 - 2x) - 2(x^2 - 3)
\][/tex]
Let's proceed step by step to simplify and factor the expression.
Step 1: Distribute the constants inside the parentheses.
[tex]\[
4(x^2 - 2x) = 4x^2 - 8x
\][/tex]
[tex]\[
-2(x^2 - 3) = -2x^2 + 6
\][/tex]
Step 2: Combine these two expressions.
[tex]\[
4x^2 - 8x - 2x^2 + 6
\][/tex]
Step 3: Simplify the combined expression by combining like terms.
[tex]\[
(4x^2 - 2x^2) + (-8x) + 6 = 2x^2 - 8x + 6
\][/tex]
Step 4: Factor the expression \(2x^2 - 8x + 6\).
First, we factor out the greatest common factor (GCF), which is 2:
[tex]\[
2(x^2 - 4x + 3)
\][/tex]
Now, we need to factor the quadratic expression inside the parentheses:
[tex]\[
x^2 - 4x + 3
\][/tex]
To factor \(x^2 - 4x + 3\), we need to find two numbers that multiply to \(3\) (the constant term) and add to \(-4\) (the coefficient of \(x\)).
These two numbers are \(-1\) and \(-3\) because:
[tex]\[
-1 \cdot (-3) = 3 \quad \text{and} \quad -1 + (-3) = -4
\][/tex]
So, the expression \(x^2 - 4x + 3\) can be factored as:
[tex]\[
(x - 1)(x - 3)
\][/tex]
Putting it all together, we have:
[tex]\[
2(x - 1)(x - 3)
\][/tex]
Thus, the factored form of the original expression is:
[tex]\[
2(x - 1)(x - 3)
\][/tex]
So, the correct answer is:
C. [tex]\(2(x-1)(x-3)\)[/tex]
