problem: For how many positive integer values of $x$ is the sum $x^2+4x+4$ less than 20?
solution: Note that since we can only use positive integers for $x$, the minimum will be x = 1.  Testing x = 2, we get $2^2 + 4\cdot 2 + 4 = 16$.  Since $3^2 - 2^2 = 5$, we know that only $x = 1,2$ will work, thus, there are $\boxed{2}$ positive integer values of $x$ such that this function is less than 20.