1) \(x^2+2x+3=\left(x+1\right)^2+2\ge2>0\)
2) \(x^2+6x+10=\left(x+3\right)^2+1\ge1>0\)
3) \(4x^2+6x+10=\left(2x+\dfrac{3}{2}\right)^2+\dfrac{31}{4}\ge\dfrac{31}{4}>0\)
4) \(3x^2+5x+3=3\left(x+\dfrac{5}{6}\right)^2+\dfrac{11}{12}\ge\dfrac{11}{12}>0\)
5) \(x^2-4x+5=\left(x-2\right)^2+1\ge1>0\)
6) \(4x^2-12x+10=\left(2x-3\right)^2+1\ge1>0\)
7) \(9x^2-6x+2=\left(3x-1\right)^2+1\ge1>0\)
8) \(2x^2-3x+5=2\left(x-\dfrac{3}{4}\right)^2+\dfrac{31}{8}\ge\dfrac{31}{8}>0\)
1: Ta có: \(x^2+2x+3\)
\(=x^2+2x+1+2\)
\(=\left(x+1\right)^2+2>0\forall x\)
2: Ta có: \(x^2+6x+10\)
\(=x^2+6x+9+1\)
\(=\left(x+3\right)^2+1>0\forall x\)
3: Ta có: \(4x^2+6x+10\)
\(=4\left(x^2+\dfrac{3}{2}x+\dfrac{5}{2}\right)\)
\(=4\left(x^2+2\cdot x\cdot\dfrac{3}{4}+\dfrac{9}{16}+\dfrac{31}{16}\right)\)
\(=4\left(x+\dfrac{3}{4}\right)^2+\dfrac{31}{4}>0\forall x\)
