## Intermediate Algebra: Connecting Concepts through Application

$\text{all real numbers}$
$\bf{\text{Solution Outline:}}$ The domain of the given rational function, $f(x)=\dfrac{x+8}{x^2+5x+21} ,$ are the values of $x$ which will NOT make the denominator equal to $0.$ $\bf{\text{Solution Details:}}$ The denominator, $x^2+5x+21,$ is equivalent to $x(x+5)+21.$ The only time the expression $x(x+5)$ becomes a negative number is when $x$ is between $-5$ to $0$ (exclusive). But with $+21,$ the denominator still becomes positive. Any other value of $x$ guarantees that the denominator is a positive real number. Hence, the denominator is always a positive real number (i.e. the denominator never becomes $0.$) Hence, the domain is the set of $\text{all real numbers} .$