Task 909

Task 909: development task in emrekuruu/math-openenv, an RL environment on the Hugging Face Hub. Let $x$ and $y$ be real numbers such that \ 4x^2 + 8xy + 5y^2 = 1.\ Let $m$ and $M$ be the minimum and maximum values of $2x^2 + 3xy + 2y^2,$ respectively. Find the product $mM.$

The task

Let $x$ and $y$ be real numbers such that \[4x^2 + 8xy + 5y^2 = 1.\]Let $m$ and $M$ be the minimum and maximum values of $2x^2 + 3xy + 2y^2,$ respectively. Find the product $mM.$