Prove that a^4+b^4+a^2b^2+x^2-xy+y^2+1>0
Help would be appreciated!
The statement is not true.
$\displaystyle \displaystyle \begin{align*} a^4 + b^4 + a^2b^2 + x^2 - x\,y + y^2 + 1 &= \left( a^2 \right)^2 + 2a^2b^2 + \left( b^2 \right)^2 + x^2 - 2\,x\,y + y^2 + 1 - a^2b^2 + x\,y \\ &= \left( a^2 + b^2 \right)^2 + \left( x^2 - y^2 \right)^2 + 1 - a^2b^2 + x\,y \end{align*}$
For this to be positive, we require $\displaystyle \displaystyle \begin{align*} x\,y - a^2b^2 > 1 \end{align*}$. There is no guarantee of this.