To find the correct factored form of the given expression \(4(x^2 - 2x) - 2(x^2 - 3)\), let's go through the simplification and factoring step by step.
### Step 1: Expand and Simplify the Expression
Given expression:
[tex]\[ 4(x^2 - 2x) - 2(x^2 - 3) \][/tex]
First, distribute the terms inside the parentheses:
[tex]\[ 4(x^2) - 4(2x) - 2(x^2) - 2(-3) \][/tex]
This simplifies to:
[tex]\[ 4x^2 - 8x - 2x^2 + 6 \][/tex]
Next, combine like terms:
[tex]\[ 4x^2 - 2x^2 - 8x + 6 \][/tex]
[tex]\[ 2x^2 - 8x + 6 \][/tex]
### Step 2: Factor the Simplified Expression
Now, we factor \(2x^2 - 8x + 6\). We look for common factors and use factoring techniques:
First, notice that all terms have a common factor of 2:
[tex]\[ 2(x^2 - 4x + 3) \][/tex]
Next, we factor the quadratic expression inside the parentheses:
[tex]\[ x^2 - 4x + 3 \][/tex]
To factor \(x^2 - 4x + 3\), we look for two numbers that multiply to 3 and add to -4. Those numbers are -1 and -3.
Therefore, \(x^2 - 4x + 3\) factors as:
[tex]\[ (x - 1)(x - 3) \][/tex]
So, putting the factor of 2 back in, the factored form of \(2(x^2 - 4x + 3)\) is:
[tex]\[ 2(x - 1)(x - 3) \][/tex]
### Step 3: Verify with Given Choices
Given the choices:
A. \(2(x-1)(x-3)\)
B. \((2x-3)(x+1)\)
C. \((2x+3)(x+1)\)
D. \(2(x+1)(x+3)\)
From our factorization, the correct expression in factored form is:
[tex]\[ 2(x - 1)(x - 3) \][/tex]
Hence, the correct answer is:
A. [tex]\(2(x-1)(x-3)\)[/tex]
