which of the following expressions is equal to $-3x^2 - 12$?

a. $(-3x - 6i)(x + 2i)$

b. $(-3x + 2i)(x - 6i)$

c. $(-3x + 6i)(x + 2i)$

d. $(-3x - 4i)(x - 3i)$

Question

which of the following expressions is equal to $-3x^2 - 12$?

a. $(-3x - 6i)(x + 2i)$

b. $(-3x + 2i)(x - 6i)$

c. $(-3x + 6i)(x + 2i)$

d. $(-3x - 4i)(x - 3i)$

Accepted Answer

# Explanation:
## Step1: Expand Option A
Using the distributive property (FOIL method):
$$
\begin{align*}
(-3x - 6i)(x + 2i)&=-3x\cdot x+(-3x)\cdot 2i-6i\cdot x - 6i\cdot 2i\
&=-3x^{2}-6xi - 6xi-12i^{2}\
&=-3x^{2}-12xi + 12\quad (\text{since } i^{2}=-1)
\end{align*}
$$
This is not equal to $-3x^{2}-12$.

## Step2: Expand Option B
$$
\begin{align*}
(-3x + 2i)(x - 6i)&=-3x\cdot x+(-3x)\cdot(-6i)+2i\cdot x-2i\cdot 6i\
&=-3x^{2}+18xi + 2xi-12i^{2}\
&=-3x^{2}+20xi + 12\quad (\text{since } i^{2}=-1)
\end{align*}
$$
This is not equal to $-3x^{2}-12$.

## Step3: Expand Option C
$$
\begin{align*}
(-3x + 6i)(x + 2i)&=-3x\cdot x+(-3x)\cdot 2i+6i\cdot x+6i\cdot 2i\
&=-3x^{2}-6xi + 6xi+12i^{2}\
&=-3x^{2}-12\quad (\text{since } i^{2}=-1\text{ and }-6xi + 6xi = 0)
\end{align*}
$$
This matches the given expression. We can check Option D for completeness.

## Step4: Expand Option D
$$
\begin{align*}
(-3x - 4i)(x - 3i)&=-3x\cdot x+(-3x)\cdot(-3i)-4i\cdot x+4i\cdot 3i\
&=-3x^{2}+9xi-4xi + 12i^{2}\
&=-3x^{2}+5xi-12\quad (\text{since } i^{2}=-1)
\end{align*}
$$
This is not equal to $-3x^{2}-12$.

# Answer:
C. $(-3x + 6i)(x + 2i)$

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QUESTION IMAGE

Question

Explanation:

Step1: Expand Option A

Step2: Expand Option B

Step3: Expand Option C

Step4: Expand Option D

Answer: