Algebraic proof of an inequality being given different constraints on 2 variables Let $x,y \in \mathbb R$ and $$(1+x+x^2)(1+y+y^2)=2x^2y^2-1$$ and assume that $2<x<y$. I want to algebraically show that $xy<16$ but  I cannot get anywhere. I have tried substitution and nothing seems to help. Graphically I know that it is try because I can simply graph and check visually but I want an actual proof if any one can offer any hints or proof thank you.
 A: $$(1+x+x^2)(1+y+y^2)=2x^2y^2-1\tag{1}$$ $2<x<y$.
Plug $x=2$ in $(1)$
$$7 \left(y^2+y+1\right)=8 y^2-1\to y=-1;\;y=8$$
$(2,8)$ is a point where the curve has its maximum because
in that point the curve $(1)$ is decreasing
$$y'=\frac{-2 x y^2+2 x y+2 x+y^2+y+1}{2 x^2 y-x^2-2 x y-x-2 y-1}$$
as $y'(2,8)=-\frac{49}{3}<0$. It can be proved that the curve is decreasing in the whole region $2<x<y$.
Thus the maximum value of $xy$ is $16$.
Hope this helps

$$...$$

A: Here is a proof which, I am conscious of that, is graphical but has the merit to transform the issue  into more natural variables for the problem at hand, i.e., by taking:
$$\begin{cases}S=x+y\\ P=xy\end{cases} \ \ \iff \ \ \begin{cases}x&=&\frac12(S-\sqrt{S^2-4P})\\y&=&\frac12(S+\sqrt{S^2-4P})\end{cases}\tag{1}$$
(think to relationship $t^2-St+P=0$).
Indeed, with this change of variables, the problem is easily shown to be equivalent (due to symmetry between variables $x$ and $y$) to:
$$\text{knowing that} \ \ -P^2+PS+S^2-P+S+2=0 \tag{2}$$
$$\text{show that} \ \  P<16$$
taking into account constraint $x>2$ (see (1)) and the  implied constraint $S^2-4P \ge 0$.
Now, have a look at fig. 1 below with coordinates axes  $(S,P)$:

Fig. 1: Only the region defined by $(S>4,P>4)$ makes sense. We have extended the "scene" in order to better understand the hyperbolic curve given by equ. (2).
Point $(S,P)$ must

*

*belong to hyperbola (in blue) accounting for constraint (2).


*be below parabola (P) (in black) with equation $4P=S^2$ and


*be above the red line accounting for condition $x>2.$
The limit point is at the intersection of the blue and red curves (small square) with ordinate $S=16$ as desired.
I am convinced that one can deduce from this graphical representation a fully analytic proof.
