Other way:

P-\(5\)=\(\dfrac{3x^2+6x+10-5x^2-10x-15}{x^2+2x+3}=\dfrac{-2x^2-4x-5}{x^2+2x+3}=\dfrac{-2\left(x^2+2x+3\right)+1}{x^2+2x+3}\)

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

\(\Rightarrow P\le\dfrac{-3}{2}+5=\dfrac{7}{2}\)

\(\Rightarrow max_P=\dfrac{7}{2}\).When x=-1