Let's analyze each of the given options to determine which one is equivalent to the polynomial [tex]\(x^2 + 8\)[/tex]:
Option A: [tex]\((x + 2\sqrt{2})(x - 2\sqrt{2})\)[/tex]
To find the expansion, we use the difference of squares formula:
[tex]\[
(a + b)(a - b) = a^2 - b^2
\][/tex]
Here, [tex]\(a = x\)[/tex] and [tex]\(b = 2\sqrt{2}\)[/tex]. So,
[tex]\[
(x + 2\sqrt{2})(x - 2\sqrt{2}) = x^2 - (2\sqrt{2})^2 = x^2 - 8
\][/tex]
which is not equivalent to [tex]\(x^2 + 8\)[/tex].
Option B: [tex]\((x + 4i)(x - 4i)\)[/tex]
Using the difference of squares formula again:
[tex]\[
(a + b)(a - b) = a^2 - b^2
\][/tex]
Here, [tex]\(a = x\)[/tex] and [tex]\(b = 4i\)[/tex]. So,
[tex]\[
(x + 4i)(x - 4i) = x^2 - (4i)^2 = x^2 - 16i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex],
[tex]\[
x^2 - 16i^2 = x^2 - 16(-1) = x^2 + 16
\][/tex]
which is not equivalent to [tex]\(x^2 + 8\)[/tex].
Option C: [tex]\((x + 2\sqrt{2})^2\)[/tex]
To expand this expression, we use the formula:
[tex]\[
(a + b)^2 = a^2 + 2ab + b^2
\][/tex]
Here, [tex]\(a = x\)[/tex] and [tex]\(b = 2\sqrt{2}\)[/tex]. So,
[tex]\[
(x + 2\sqrt{2})^2 = x^2 + 2(x)(2\sqrt{2}) + (2\sqrt{2})^2 = x^2 + 4x\sqrt{2} + 8
\][/tex]
which is not equivalent to [tex]\(x^2 + 8\)[/tex].
Option D: [tex]\((x + 2\sqrt{2}i) \cdot (x - 2\sqrt{2}i)\)[/tex]
Using the difference of squares formula one more time:
[tex]\[
(a + b)(a - b) = a^2 - b^2
\][/tex]
Here, [tex]\(a = x\)[/tex] and [tex]\(b = 2\sqrt{2}i\)[/tex]. So,
[tex]\[
(x + 2\sqrt{2}i) \cdot (x - 2\sqrt{2}i) = x^2 - (2\sqrt{2}i)^2 = x^2 - 8(i^2)
\][/tex]
Since [tex]\(i^2 = -1\)[/tex],
[tex]\[
x^2 - 8(i^2) = x^2 - 8(-1) = x^2 + 8
\][/tex]
This matches the given polynomial [tex]\(x^2 + 8\)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{4}
\][/tex]
