select the correct answer. which expression is equivalent to this polynomial? $x^{2}+12$ a. $(x +…

select the correct answer. which expression is equivalent to this polynomial? $x^{2}+12$ a. $(x + 2sqrt{3}i)(x - 2sqrt{3}i)$ b. $(x + 6i)(x - 6i)$ c. $(x + 2sqrt{3})^{2}$ d. $(x + 2sqrt{3})(x - 2sqrt{3})$

select the correct answer. which expression is equivalent to this polynomial? $x^{2}+12$ a. $(x + 2sqrt{3}i)(x - 2sqrt{3}i)$ b. $(x + 6i)(x - 6i)$ c. $(x + 2sqrt{3})^{2}$ d. $(x + 2sqrt{3})(x - 2sqrt{3})$

Answer

Explanation:

Step1: Recall the difference - of - squares formula

The difference - of - squares formula is $(a + b)(a - b)=a^{2}-b^{2}$. We want to rewrite $x^{2}+12$ in a factored form. We know that $x^{2}+12=x^{2}-(- 12)$.

Step2: Express $-12$ as a square

Since $-12=(2\sqrt{3}i)^{2}$ (because $i^{2}=-1$ and $(2\sqrt{3})^{2}=12$), then $x^{2}+12=x^{2}-(2\sqrt{3}i)^{2}$.

Step3: Apply the difference - of - squares formula

Using the formula $(a + b)(a - b)=a^{2}-b^{2}$ with $a = x$ and $b = 2\sqrt{3}i$, we get $x^{2}-(2\sqrt{3}i)^{2}=(x + 2\sqrt{3}i)(x - 2\sqrt{3}i)$.

Answer:

A. $(x + 2\sqrt{3}i)(x - 2\sqrt{3}i)$