problem: What is the smallest integer value of $c$ such that the function $f(x)=\frac{2x^2+x+5}{x^2+4x+c}$ has a domain of all real numbers?
solution: The given function has a domain of all real numbers if and only if the denominator is never equal to zero.  In other words, the quadratic $x^2 + 4x + c = 0$ has no real roots.  The discriminant of this quadratic is $16 - 4c$.  The quadratic has no real roots if and only if the discriminant is negative, so $16 - 4c < 0$, or $c > 4$.  The smallest integer $c$ that satisfies this inequality is $c = \boxed{5}$.