To determine which expression is equivalent to the polynomial [tex]\(x^2 + 12\)[/tex], let's factor it completely.
We start by using the difference of squares identity: [tex]\(a^2 - b^2 = (a+b)(a-b)\)[/tex]. However, since our polynomial is [tex]\(x^2 + 12\)[/tex], we need to incorporate complex numbers, as the sum of squares can be factored in the form involving complex numbers:
[tex]\[ x^2 + a^2 = (x + ai)(x - ai) \][/tex]
In this case, we can rewrite 12 in the form of [tex]\(a^2\)[/tex] where [tex]\(a = 2\sqrt{3}\)[/tex]. Therefore, [tex]\(12 = (2\sqrt{3})^2\)[/tex]:
Now, our polynomial becomes:
[tex]\[ x^2 + (2\sqrt{3})^2 \][/tex]
Using the complex number factorization, we have:
[tex]\[ x^2 + (2\sqrt{3})^2 = (x + 2\sqrt{3}i)(x - 2\sqrt{3}i) \][/tex]
Thus, the equivalent expression is:
[tex]\[ (x + 2\sqrt{3}i)(x - 2\sqrt{3}i) \][/tex]
Comparing this with the given options, we see that Option A is:
[tex]\[ (x + 2\sqrt{3}i)(x - 2\sqrt{3}i) \][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{1}
\][/tex]
