Answer
Domain = $\mathbb{R}\backslash\{-1,-7\}$ or $\{x\in \mathbb{R}\ |\ x\neq-1$ and $ x\neq-7\}$
Work Step by Step
The domain of a real function ( a function defined on real numbers)
is the set of numbers for which f(x) is defined.
$ f(x)=\displaystyle \frac{2x-7}{x^{2}+8x+7}\quad$ can be calculated for any real number,
except for those yielding a zero in the denominator.
Here,
$ x^{2}+8x+7=0\quad$ ... factor: two factors of 7 with sum 8 ...
$(x+1)(x+7)=0$
$x=-1$ and $x=-7\quad $must be excluded from the domain.
Domain = $\mathbb{R}\backslash\{-1,-7\}$ or $\{x\in \mathbb{R}\ |\ x\neq-1,$ and$ x\neq-7\}$
