College Algebra, Chapter 1, 1.4, Section 1.4, Problem 52

Evaluate $\displaystyle (\sqrt{3} - \sqrt{-4})(\sqrt{6} - \sqrt{-8})$ and express the result in the form $a + bi$.


$
\begin{equation}
\begin{aligned}

=& (\sqrt{3} - \sqrt{-4})(\sqrt{6} - \sqrt{-8})
&& \text{Given}
\\
\\
=& \sqrt{3\cdot6} - \sqrt{3(-8)} - \sqrt{6(-4)} + \sqrt{(-8)(-4)}
&& \text{Use FOIL method}
\\
\\
=& \sqrt{18} - \sqrt{-24} - \sqrt{-24} + \sqrt{32}
&& \text{Simplify}
\\
\\
=& \sqrt{18} - 2 \sqrt{-24} + \sqrt{32}
&& \text{Recall that } i^2 = -1
\\
\\
=& \sqrt{18} - 2 \sqrt{24i^2} + \sqrt{32}
&&
\\
\\
=& \sqrt{18} - 2 \sqrt{24} i + \sqrt{32}
&& \text{Group terms}
\\
\\
=& (\sqrt{18} + \sqrt{32}) - 2 \sqrt{24} i
&& \text{Simplify}
\\
\\
=& 7 \sqrt{2} - 4 \sqrt{6} i
&&

\end{aligned}
\end{equation}
$