Simplify each of the following and express it in the form a + ib

Given: $(-2+\sqrt{-3})(-3+2 \sqrt{-3})$

We re – write the above equation

$(-2+\sqrt{(-1) \times 3})(-3+2 \sqrt{(-1) \times 3})$

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

$\left[\because, i^{2}=-1\right]$

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

Now, open the brackets

$=-2 \times(-3)+(-2) \times 2 \mathrm{i} \sqrt{3}+\mathrm{i} \sqrt{3} \times(-3)+\mathrm{i} \sqrt{3} \times 2 \mathrm{i} \sqrt{3}$

$=6-4 \mathrm{i} \sqrt{3}-3 \mathrm{i} \sqrt{3}+6 \mathrm{i}^{2}$

$=6-7 \mathrm{i} \sqrt{3}+[6 \times(-1)]\left[\because, \mathrm{i}^{2}=-1\right]$

$=6-7 \mathrm{i} \sqrt{3}-6$

$=0-7 \mathrm{i} \sqrt{3}$