To factor the expression [tex]\(2x^2 + 8\)[/tex], we start by looking for any common factors.
1. Identify the common factor:
The terms [tex]\(2x^2\)[/tex] and [tex]\(8\)[/tex] have a common factor of [tex]\(2\)[/tex]. So, we can factor out [tex]\(2\)[/tex] from the expression:
[tex]\[
2x^2 + 8 = 2(x^2 + 4)
\][/tex]
2. Analyze the remaining quadratic expression:
Next, we need to see if [tex]\(x^2 + 4\)[/tex] can be factored further. The expression [tex]\(x^2 + 4\)[/tex] does not have any real roots because it equates to zero would mean solving [tex]\(x^2 = -4\)[/tex], which does not have real solutions. Therefore, [tex]\(x^2 + 4\)[/tex] is not factorable into simpler expressions involving real numbers.
3. Simplification and comparison:
So, the factored form of the original expression, using real numbers, is:
[tex]\[
2(x^2 + 4)
\][/tex]
Our goal now is to match this with one of the provided choices.
Let's review each option:
- Option (A): [tex]\((2x + 4)(2x - 4)\)[/tex]
Multiplying [tex]\((2x + 4)(2x - 4)\)[/tex] yields:
[tex]\[
(2x + 4)(2x - 4) = 4x^2 - 16
\][/tex]
This is not equal to [tex]\(2x^2 + 8\)[/tex], so option (A) is incorrect.
- Option (B): Prime
The term "prime" indicates that the expression cannot be factored further. However, we have already factored out a common term [tex]\(2\)[/tex], so the expression is not prime.
- Option (C): [tex]\(2(x + 2)(x - 2)\)[/tex]
Multiplying [tex]\(2(x + 2)(x - 2)\)[/tex] yields:
[tex]\[
2(x + 2)(x - 2) = 2(x^2 - 4) = 2x^2 - 8
\][/tex]
This is not equal to [tex]\(2x^2 + 8\)[/tex], so option (C) is incorrect.
- Option (D): None of these
The form [tex]\(2(x^2 + 4)\)[/tex] isn't listed among the provided options (A), (B), or (C). Thus, the correct response is option (D), "None of these."
Thus, the correct answer is
[tex]\[
(D) \text{None of these}
\][/tex]
