\(P\left(x\right)+\left(x^3+2x^2+5x+\dfrac{1}{2}\right)=3x^3+3x^2+5x-\dfrac{1}{2}\)
\(\Rightarrow P\left(x\right)=3x^3+3x^2+5x-\dfrac{1}{2}-\left(x^3+2x^2+5x+\dfrac{1}{2}\right)\)
\(\Rightarrow P\left(x\right)=2x^3+x^2-1\)
\(P\left(x\right)=3x^3+3x^2+5x-\dfrac{1}{2}-\left(x^3+2x^2+5x+\dfrac{1}{2}\right)\)
\(P\left(x\right)=3x^3+3x^2+5x-\dfrac{1}{2}-x^3-2x^2-5x-\dfrac{1}{2}\)
\(P\left(x\right)=2x^3+x^2-1\)