\(A=x^2-2.\dfrac{3}{2}x+\dfrac{9}{4}+\dfrac{31}{4}=\left(x-\dfrac{3}{2}\right)^2+\dfrac{31}{4}\)
Do \(\left\{{}\begin{matrix}\left(x-\dfrac{3}{2}\right)^2\ge0;\forall x\\\dfrac{31}{4}>0\end{matrix}\right.\) \(\Rightarrow A>0;\forall x\)
\(B=-\left(4x^2+4x+1\right)-\left(9y^2-6y+1\right)-8\)
\(=-\left(2x+1\right)^2-\left(3y-1\right)^2-8\)
Do \(\left\{{}\begin{matrix}-\left(2x+1\right)^2\le0;\forall x\\-\left(3y-1\right)^2\le0;\forall y\\-8< 0\end{matrix}\right.\) \(\Rightarrow B< 0;\forall x,y\)
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