Simplify (8+ square root of -12)-(4+2 square root of 3i)

$(8 + \sqrt{- 12}) - (4 + 2 \sqrt{3} i)$

Simplify each term.

Rewrite $- 12$ as $- 1 (12)$ .

$8 + \sqrt{- 1 (12)} - (4 + 2 \sqrt{3} i)$

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$8 + \sqrt{- 1} \cdot \sqrt{12} - (4 + 2 \sqrt{3} i)$

Rewrite $\sqrt{- 1}$ as $i$ .

$8 + i \cdot \sqrt{12} - (4 + 2 \sqrt{3} i)$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$8 + i \cdot \sqrt{4 (3)} - (4 + 2 \sqrt{3} i)$

Rewrite $4$ as $2^{2}$ .

$8 + i \cdot \sqrt{2^{2} \cdot 3} - (4 + 2 \sqrt{3} i)$

Pull terms out from under the radical.

$8 + i \cdot (2 \sqrt{3}) - (4 + 2 \sqrt{3} i)$

Move $2$ to the left of $i$ .

$8 + 2 \cdot i \sqrt{3} - (4 + 2 \sqrt{3} i)$

Apply the distributive property.

$8 + 2 i \sqrt{3} - 1 \cdot 4 - (2 \sqrt{3} i)$

Multiply $- 1$ by $4$ .

$8 + 2 i \sqrt{3} - 4 - (2 \sqrt{3} i)$

Multiply $2$ by $- 1$ .

$8 + 2 i \sqrt{3} - 4 - 2 \sqrt{3} i$

Simplify by adding terms.

Subtract $4$ from $8$ .

$2 i \sqrt{3} + 4 - 2 \sqrt{3} i$

Reorder the factors of $- 2 \sqrt{3} i$ .

$2 i \sqrt{3} + 4 - 2 i \sqrt{3}$

Subtract $2 i \sqrt{3}$ from $2 i \sqrt{3}$ .

$0 + 4$

Add $0$ and $4$ .

$4$