To determine which expression is equivalent to the polynomial \(x^2 + 12\), we need to consider how this polynomial can be factored, especially focusing on complex numbers since the given polynomial has a positive constant term added to \(x^2\).
Given the polynomial:
[tex]\[ x^2 + 12, \][/tex]
we want to find two binomials whose product is this polynomial. We'll examine the provided options one by one.
Option A: \((x + 2\sqrt{3}i)(x - 2\sqrt{3}i)\)
Let's expand this:
[tex]\[ (x + 2\sqrt{3}i)(x - 2\sqrt{3}i) = x^2 - (2\sqrt{3}i)^2 = x^2 - 4 \cdot 3 \cdot (-1) = x^2 + 12. \][/tex]
This matches the given polynomial.
Option B: \((x + 6i)(x - 6i)\)
Let's expand this:
[tex]\[ (x + 6i)(x - 6i) = x^2 - (6i)^2 = x^2 - 36(-1) = x^2 + 36. \][/tex]
This does not match the given polynomial.
Option C: \((x + 2\sqrt{3})^2\)
Let's expand this:
[tex]\[ (x + 2\sqrt{3})^2 = x^2 + 2 \cdot x \cdot 2\sqrt{3} + (2\sqrt{3})^2 = x^2 + 4\sqrt{3}x + 12. \][/tex]
This does not match the given polynomial.
Option D: \((x + 2\sqrt{3})(x - 2\sqrt{3})\)
Let's expand this:
[tex]\[ (x + 2\sqrt{3})(x - 2\sqrt{3}) = x^2 - (2\sqrt{3})^2 = x^2 - 4 \cdot 3 = x^2 - 12. \][/tex]
This does not match the given polynomial.
Based on the expansions and matchings with the polynomial \(x^2 + 12\), the correct answer is:
[tex]\[ \boxed{B} \][/tex]
