<s>[INST] For what $x$ and $y$ polynomial has maximum value? 
For what $x,y\in\mathbb R$ does the polynomial
  $$-5x^2-2xy-2y^2+14x+10y-1$$
  attain a maximum?

My attempt:
I called $\alpha$ maximum value.
$$-5x^2-2xy-2y^2+14x+10y-1\leqslant\alpha$$
$$-5x^2-2xy-2y^2+14x+10y-1-\alpha\leqslant 0$$
$$5x^2+2xy+2y^2-14x-10y+1+\alpha\geqslant 0$$
$$(x+y)^2+(y-5)^2+3x^2+(x-7)^2-73+\alpha\geqslant0$$
$$\alpha\geqslant73$$
So the lowest maximum value turned out to be $73$, but after checking answers I was wrong-maximum is $16$, so my further plans to calculate from that $x$ and $y$ seemed purposless. I'd like to see solution using only high school knowledge.
Ans: $x=1$, $y=2$
 [/INST] Your idea is quite correct, but you have to complete squares in a different way:
\begin{gather}
-(5x^2+2xy+2y^2-14x-10y+1) =\\
-\left[\left(x\sqrt 5 + y\frac{1}{\sqrt 5} -\frac{7}{\sqrt 5}\right)^2 +2y^2-10y+1-\frac{1}{5}y^2-\frac{49}{5}+\frac{14}{5}y\right]=\\
-\left[\left(x\sqrt 5 + y\frac{1}{\sqrt 5} -\frac{7}{\sqrt 5}\right)^2 +\frac{9}{5}y^2-\frac{36}{5}y-\frac{44}{5}\right]=\\
-\left[\left(x\sqrt 5 + y\frac{1}{\sqrt 5} -\frac{7}{\sqrt 5}\right)^2 +\left(\frac{3}{\sqrt 5}y-\frac{6}{\sqrt 5}\right)^2-\frac{36}{5}-\frac{44}{5}\right]=\\
-\left[\left(x\sqrt 5 + y\frac{1}{\sqrt 5} -\frac{7}{\sqrt 5}\right)^2 +\left(\frac{3}{\sqrt 5}y-\frac{6}{\sqrt 5}\right)^2-16\right]=\\
-\left(x\sqrt 5 + y\frac{1}{\sqrt 5} -\frac{7}{\sqrt 5}\right)^2 -\left(\frac{3}{\sqrt 5}y-\frac{6}{\sqrt 5}\right)^2+16 \leq 16
\end{gather}
In the first passage I complete the squares in order to eliminate the terms in $x^2$ and $xy$. Then complete in order to eliminate the term in $y^2$ and $y$. So we showed that our function is less or equal to $16$. Imposing the two squares equal to zero you obtain the maximum $16$ and if you do the computation you obtain the point $(1,2)$ that makes it.
This is a standard method to do this kind of exercise. You have to complete squares in a similar way as I did; then you obtain a inequality and putting the squares equal to zero you will find the minimum and the point that makes it.
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