To determine which expression is equivalent to the polynomial [tex]\( x^2 + 12 \)[/tex], let's expand each option and compare it to [tex]\( x^2 + 12 \)[/tex].
### Option A: [tex]\((x + 2\sqrt{3}i)(x - 2\sqrt{3}i)\)[/tex]
To expand this expression, we use the difference of squares formula:
[tex]\[
(x + 2\sqrt{3}i)(x - 2\sqrt{3}i) = x^2 - (2\sqrt{3}i)^2
\][/tex]
Next, calculate [tex]\((2\sqrt{3}i)^2\)[/tex]:
[tex]\[
(2\sqrt{3}i)^2 = 4 \cdot 3 \cdot i^2 = 12 \cdot (-1) = -12
\][/tex]
Thus:
[tex]\[
x^2 - (-12) = x^2 + 12
\][/tex]
This matches the given polynomial [tex]\( x^2 + 12 \)[/tex].
### Option B: [tex]\((x + 6i)(x - 6i)\)[/tex]
Using the same difference of squares formula:
[tex]\[
(x + 6i)(x - 6i) = x^2 - (6i)^2
\][/tex]
Calculate [tex]\((6i)^2\)[/tex]:
[tex]\[
(6i)^2 = 36i^2 = 36 \cdot (-1) = -36
\][/tex]
Thus:
[tex]\[
x^2 - (-36) = x^2 + 36
\][/tex]
This does not match the given polynomial [tex]\( x^2 + 12 \)[/tex].
### Option C: [tex]\((x + 2\sqrt{5})^2\)[/tex]
Expand this by squaring the binomial:
[tex]\[
(x + 2\sqrt{5})^2 = x^2 + 2 \cdot 2\sqrt{5} \cdot x + (2\sqrt{5})^2
\][/tex]
Calculate:
[tex]\[
(2\sqrt{5})^2 = 4 \cdot 5 = 20
\][/tex]
Thus:
[tex]\[
x^2 + 4\sqrt{5}x + 20
\][/tex]
This does not match the given polynomial [tex]\( x^2 + 12 \)[/tex].
### Option D: [tex]\((x + 2\sqrt{3})(x - 2\sqrt{3})\)[/tex]
Again, using the difference of squares formula:
[tex]\[
(x + 2\sqrt{3})(x - 2\sqrt{3}) = x^2 - (2\sqrt{3})^2
\][/tex]
Calculate [tex]\((2\sqrt{3})^2\)[/tex]:
[tex]\[
(2\sqrt{3})^2 = 4 \cdot 3 = 12
\][/tex]
Thus:
[tex]\[
x^2 - 12
\][/tex]
This does not match the given polynomial [tex]\( x^2 + 12 \)[/tex].
### Conclusion
After expanding all the options and comparing them with [tex]\( x^2 + 12 \)[/tex], we find that the correct answer is:
[tex]\[
\boxed{(x + 2\sqrt{3}i)(x - 2\sqrt{3}i)}
\][/tex]
So, the correct option is A.
