Select the correct answer.

Which expression in factored form is equivalent to this expression?

[tex]\[ 4(x^2 - 2x) - 2(x^2 - 3) \][/tex]

A. \(2(x-1)(x-3)\)

B. \((2x-3)(x+1)\)

C. \((2x+3)(x+1)\)

D. [tex]\(2(x+1)(x+3)\)[/tex]



Answer :

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]