Hi everyone! I have a question about how to find the closed form of a function defined by

$$\phi(\theta)=\inf_{x\geq 2}f(x;\theta)\equiv\inf_{x\geq 2}\frac{(x+2)^2}{\frac{1}{\theta}\left(\frac{x-1}{2}-\frac{1}{x}\right)+\frac{x^2-1}{16}},\ \ \theta>0$$

Since finding the minimum of $f(x;\theta)$ leads to solving a cubic polynomial. Could someone help me to characterize $\phi(\theta)$ please? Many thanks!