For $x$ > 0, the least value of $(x^2+4x+9)/x$ is:

Divide term by term: $(x^2+4x+9)/x=x+4+9/x$. By AM ≥ GM the positive pair $x$ and $9/x$ satisfies $x+9/x$ ≥ 2√9 = 6, with equality at $x=3$, so the whole expression is at least 4 + 6 = 10. (6 is that bound alone, before the constant 4 is added; 14 is the value at $x=1$; 9 is the constant term of the numerator.) At $x=3$ the value is $(9+12+9)/3=10$, while $x=2$ gives 21/2 = 10.5.