Show that the equation

$2{\sec }^{2}x+4{\tan }^{2}x+10\sec x=0$

can be written in the form

$a{\sec }^{2}x+b\sec x+c=0$

where $a$, $b$ and $c$ are integers with $a>0$ and with no common factor other than 1.

Hence, given that $x$ is obtuse, find the exact value of $\tan x$.