jmc

We see that $x^2+4x+4=(x+2{)}^2$. If $x$ must be positive, we can see that this expression can take on the value of any perfect square that is greater than or equal to $(1+2{)}^2=9$. The possible values between 10 and 50 are thus 16, 25, 36, and 49, achieved when $x=2,3,4,5$ respectively. So, there are $\boxed{4}$ positive integers $x$ for which $x^2+4x+4$ is between 10 and 50.