Question 374965
<font face="Garamond" size="+2">


There are a couple of ways to skin this cat.  The first way is just to use FOIL and go at it straight on.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left(5\sqrt{6}\ -\ 15\sqrt{3}\right)\left(4\sqrt{6}\ -\ 12\sqrt{3}\right)\ =\ 20\,\cdot\,6\ +\ 60\sqrt{18}\ -\ 60\sqrt{18}\ -\ 180\,\cdot\,3\ =\ -420]


Or notice that the first binomial has a 5 that you can factor out and the second binomial has a 4 that you can factor out:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 5\,\cdot\,\left(\sqrt{6}\ -\ 3\sqrt{3}\right)\,\cdot\,4\,\cdot\,\left(\sqrt{6}\ -\ 3\sqrt{3}\right)]


Now notice that the remaining binomial factors are a conjugate pair, hence the product is the difference of two squares:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 5\ \cdot\ 4\ \left(\sqrt{6}\ -\ 3\sqrt{3}\right)\left(\sqrt{6}\ +\ 3\sqrt{3}\right)\ =\ 20\ \cdot\ (6\ -\ 27)\ =\ -420]



John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
<div style="text-align:center"><a href="http://outcampaign.org/" target="_blank"><img src="http://cdn.cloudfiles.mosso.com/c116811/scarlet_A.png" border="0" alt="The Out Campaign: Scarlet Letter of Atheism" width="143" height="122" /></a></div>
</font>