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 $a + bi$.


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