\(P=\dfrac{3x^2+6x+11}{x^2+2x+3}\)
\(P=\dfrac{4x^2+8x+12-x^2-2x-1}{x^2+2x+3}\)
\(P=\dfrac{4\left(x^2+2x+3\right)}{x^2+2x+3}-\dfrac{\left(x+1\right)^2}{\left(x+1\right)^2+2}\)
\(P=4-\dfrac{\left(x+1\right)^2}{\left(x+1\right)^2+2}\)
Do : \(-\dfrac{\left(x+1\right)^2}{\left(x+1\right)^2+2}\) ≤ 0 ∀x
⇒ \(4-\dfrac{\left(x+1\right)^2}{\left(x+1\right)^2+2}\) ≤ 4
⇒ PMax = 4 ⇔ x = - 1
\(P=\dfrac{3x^2+6x+11}{x^2+2x+3}=\dfrac{3x^2+6x+9+2}{x^2+2x+3}=\dfrac{3\left(x^2+2x+3\right)+2}{x^2+2x+3}=3+\dfrac{2}{x^2+2x+3}=3+\dfrac{2}{\left(x+1\right)^2+2}\le3+1=4\)
