CM : 2X2+13y2-10xy-4x+12y+5 > 0
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\(a,\Leftrightarrow\left(x^2-2xy+y^2\right)+\left(x^2+x+\dfrac{1}{4}\right)+\dfrac{7}{4}=0\\ \Leftrightarrow\left(x-y\right)^2+\left(x+\dfrac{1}{2}\right)^2+\dfrac{7}{4}=0\\ \Leftrightarrow x,y\in\varnothing\left[\left(x-y\right)^2+\left(x+\dfrac{1}{2}\right)^2+\dfrac{7}{4}\ge\dfrac{7}{4}>0\right]\\ b,\Leftrightarrow\left(x^2-2x+1\right)+\left(9y^2+12y+4\right)+\left(4z^2-4z+1\right)+14=0\\ \Leftrightarrow\left(x-1\right)^2+\left(3y+2\right)^2+\left(2z-1\right)^2+14=0\\ \Leftrightarrow x,y,z\in\varnothing\left[\left(x-1\right)^2+\left(3y+2\right)^2+\left(2z-1\right)^2+14\ge14>0\right]\)
\(c,\Leftrightarrow-\left(x^2-10xy+25y^2\right)-\left(y^2-20y+100\right)-50=0\\ \Leftrightarrow-\left(x-5y\right)^2-\left(y-10\right)^2-50=0\\ \Leftrightarrow x,y\in\varnothing\left[-\left(x-5y\right)^2-\left(y-10\right)^2-50\le-50< 0\right]\)
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\(T=-2\left(x^2+y^2+1-2xy+2x-2y\right)-2y^2+8y+2004\)
\(T=-2\left(x-y+1\right)^2-2\left(y-2\right)^2+2012\le2012\)
\(T_{max}=2012\) khi \(\left\{{}\begin{matrix}x=1\\y=2\end{matrix}\right.\)
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\(x^3+2x^2-2x-1\)
\(=\left(x-1\right)\left(x^2+x+1\right)+2x\left(x-1\right)\)
\(=\left(x-1\right)\left(x^2+3x+1\right)\)
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11: \(2x^2-12xy+18y^2\)
\(=2\left(x^2-6xy+9y^2\right)\)
\(=2\left(x-3y\right)^2\)
12: \(\left(x^2+x\right)^2+3\left(x^2+x\right)+2\)
\(=\left(x^2+x+2\right)\left(x^2+x+1\right)\)
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\(2x^2+13y^2-10xy-4x+12y+5\)
\(=\frac{1}{2}\left[4x^2+26y^2-20xy-8x+24y+10\right]\)
\(=\frac{1}{2}\left[\left(4x^2+25y^2+4-20xy+20y-8x\right)+y^2+4y+4+2\right]\)
\(=\frac{1}{2}\left[\left(2x\right)^2+\left(5y\right)^2+2^2-2.2x.5y+2.5y.2-2.2x.2+\left(y+2\right)^2+2\right]\)
\(=\frac{1}{2}\left[\left(2x-5y-2\right)^2+\left(y+2\right)^2+2\right]>0\forall x;y\)