I need help showing that
$$f(x,y) := \frac{2 (4 x+5)k}{\left(2 x^2+5 x+y+4\right)^3}+\frac{-4 x-5}{\left(2 x^2+5 x+y+4\right)^2}+\frac{2}{(x+y+2)^2}\geq 0$$
with $x\geq0$, $y\geq0$, and $\frac{2 x^2+3 x+2}{2 x+1}>y$. Here $k\geq0$ is a free variable for us to choose, i.e., if we can say that for $k = 1$, the function is always non-negative, that will be great.
I've tried differentiating but $f_y$ is not always negative, so I can't substitute $y=\frac{2 x^2+3 x+2}{2 x+1}$ and show that that is always positive for some values of $k$.
I'm wondering if I could write it as an optimization problem:
$$\begin{array}{ll} \text{minimize} & f(x,y)\\ \text{subject to} & x, y \geq 0\\ & \dfrac{2 x^2+3 x+2}{2 x+1} > y\end{array}$$
and show that the minimal value is greater than $0$. Are there any algorithms to solve this? It's probably not convex.