Question

CMR với mọi giá trị của biến x ta luôn có:

(x² + 2x + 3)(x² + 2x + 4) + 3 > 0

Answer 1

Ta có : \(\left(x^2+2x+3\right)\left(x^2+2x+4\right)+3\)

\(=\left(x^2+2x+\dfrac{7}{2}-\dfrac{1}{2}\right)\left(x^2+2x+\dfrac{7}{2}+\dfrac{1}{2}\right)+3\)

\(=\left(x^2+2x+\dfrac{7}{2}\right)^2-\dfrac{1}{4}+3\)

\(=\left(x^2+2x+\dfrac{7}{2}\right)^2+\dfrac{11}{4}\)

Do \(\left(x^2+2x+\dfrac{7}{2}\right)^2\ge0\forall x\)

\(\Rightarrow\left(x^2+2x+\dfrac{7}{2}\right)^2+\dfrac{11}{4}\ge\dfrac{11}{4}>0\forall x\)

\(\Rightarrow\left(x^2+2x+3\right)\left(x^2+2x+4\right)+3>0\forall x\)

\(\left(đpcm\right)\)

:D