Intermediate Algebra: Connecting Concepts through Application

$\text{all real numbers}$
$\bf{\text{Solution Outline:}}$ The domain of the given rational function, $f(x)=\dfrac{2x+1}{x^2+3x+19} ,$ are the values of $x$ which will NOT make the denominator equal to $0.$ $\bf{\text{Solution Details:}}$ The denominator, $x^2+3x+19,$ is equivalent to $x(x+3)+19.$ By substituting any real value for $x,$ the expression $x(x+3)+19$ is always positive. Hence, the denominator is always a positive real number (i.e. the denominator never becomes $0.$) Hence, the domain is the set of $\text{all real numbers} .$