$$ ( x - 3 ) ( x + 3 - \sqrt { 3 } i ) ( x + 3 + \frac { \sqrt { 3 } i } { - } i ) $$

Question

UpStudy AI · Accepted Answer

Let's simplify and expand the given expression:

$$
(x - 3)(x + 3 - \sqrt{3}i)\left(x + 3 + \frac{\sqrt{3}i}{-}i\right)
$$

First, let's interpret the third factor correctly. It appears there might be a typo, so we'll assume it should be:

$$
(x + 3 + \sqrt{3}i)
$$

Now, the expression becomes:

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

### Step 1: Multiply the Second and Third Factors

Notice that $$(x + 3 - \sqrt{3}i)$$ and $$(x + 3 + \sqrt{3}i)$$ are complex conjugates. Multiplying them gives:

$$
(x + 3 - \sqrt{3}i)(x + 3 + \sqrt{3}i) = (x + 3)^2 - (\sqrt{3}i)^2
$$

Calculate each part:

$$
(x + 3)^2 = x^2 + 6x + 9
$$
$$
(\sqrt{3}i)^2 = 3i^2 = 3(-1) = -3
$$

So,

$$
(x + 3 - \sqrt{3}i)(x + 3 + \sqrt{3}i) = x^2 + 6x + 9 + 3 = x^2 + 6x + 12
$$

### Step 2: Multiply by the First Factor

Now, multiply the result by $$(x - 3)$$:

$$
(x - 3)(x^2 + 6x + 12) = x(x^2 + 6x + 12) - 3(x^2 + 6x + 12)
$$

Expand each term:

$$
= x^3 + 6x^2 + 12x - 3x^2 - 18x - 36
$$

Combine like terms:

$$
= x^3 + (6x^2 - 3x^2) + (12x - 18x) - 36
$$
$$
= x^3 + 3x^2 - 6x - 36
$$

### Final Answer

$$
x^3 + 3x^2 - 6x - 36
$$
The simplified expression is $$ x^3 + 3x^2 - 6x - 36 $$.

\( ( x - 3 ) ( x + 3 - \sqrt { 3 } i ) ( x + 3 + \frac { \sqrt { 3 } i } { - } i ) \)