\(2.\) \(x^2+3y^2+2xy-10x-14y+10=0\\\)
\(\Leftrightarrow x^2+2xy+y^2+2y^2-10x-14y+10=0\)
\(\Leftrightarrow\left(x+y\right)^2-2\left(x+y\right).5+25+2y^2-4y+2=17\)
\(\Leftrightarrow\left(x+y-5\right)^2+2\left(y-1\right)^2=17\)
\(\Leftrightarrow-\sqrt{17}\le x+y-5\le\sqrt{17}\Leftrightarrow5-\sqrt{17}\le x+y\le5+\sqrt{15}\)
\(3;\) \(\dfrac{3x^2}{2}+y^2+z^2+yz=1\)
\(\Leftrightarrow3x^2+2y^2+2z^2+2yz=2\Leftrightarrow\left(x+y+z\right)^2+\left(x-z\right)^2+\left(x-y^2\right)=2\Leftrightarrow-\sqrt{2}\le x+y+z\le\sqrt{2}\)
\(1;\) \(P=x+y+1\Rightarrow x=P-y-1\)
\(\Rightarrow\left(P-y-1\right)^2+3y^2+2y\left(P-y-1\right)+7\left(P-y-1+y\right)+2y^2+10=0\)
\(\Leftrightarrow\left(P-y-1\right)^2+3y^2+2yP-2y+7P-7+10=0\)
\(\Leftrightarrow P^2+4y^2+2y\left(P-1\right)+7P+4=0\)
\(\Delta'=\left(P-1\right)^2-4\left(P^2+7P+4\right)\ge0\)
\(\Leftrightarrow-3P^2-30P-15\ge0\Leftrightarrow-5-2\sqrt{5}\le P\le5+2\sqrt{5}\)
\(\Rightarrow-5-2\sqrt{5}\le x+y+1\le5+2\sqrt{5}\)
