The problem requires us to add two complex numbers: $(4 + i\sqrt{3})$ and $(-6 - 2i\sqrt{3})$.

1. Problem Description

The problem requires us to add two complex numbers: $(4 + i\sqrt{3})$ and $(-6 - 2i\sqrt{3})$.

2. Solution Steps

We add complex numbers by adding their real parts and their imaginary parts separately.

$(a + bi) + (c + di) = (a+c) + (b+d)i$

In our case:

$a = 4$, $b = \sqrt{3}$, $c = -6$, $d = -2\sqrt{3}$.

So, we have:

$(4 + i\sqrt{3}) + (-6 - 2i\sqrt{3}) = (4 - 6) + (\sqrt{3} - 2\sqrt{3})i$

$4 - 6 = -2$

$\sqrt{3} - 2\sqrt{3} = -1\sqrt{3} = -\sqrt{3}$

Therefore,

$(4 + i\sqrt{3}) + (-6 - 2i\sqrt{3}) = -2 - i\sqrt{3}$

3. Final Answer

$-2 - i\sqrt{3}$