Question

log (10+|x+6|/-x^2-4x-10) find the domain

-x^2-4x (-2)^2>10
x>-2+ or -square root 14

10+|x+6|>0
x+6>-10
x>-16
x+6>10
x>4

Answer 1

For the inside argument, the domain is all real numbers except where the denominator is zero.

If your denominator is
-x^2-4x-10
it is never zero, since the discriminant is 16-40 < 0

So, the domain is (-∞,∞)

Now, for the log to be defined, we need

10+|x+6|/(-x^2-4x-10) > 0
|x+6|/(-x^2-4x-10) > -10
It should not take you too long to see that this is always true.

So, the domain of the entire function is (-∞,∞)

The graph confirms this at

http://www.wolframalpha.com/input/?i=10%2B%7Cx%2B6%7C%2F(-x%5E2-4x-10)+for+-10+%3C%3D+x+%3C%3D+10