Ta có:
\(P\left(0\right)=a.0^2+b.0+c=-2\Rightarrow c=-2\)
\(4P\left(x\right)-P\left(2x-1\right)=6x-6\)
\(\Leftrightarrow4\left(ax^2+bx+c\right)-a\left(2x-1\right)^2-b\left(2x-1\right)-c=6x-6\)
\(\Leftrightarrow4ax^2+4bx+4c-a\left(4x^2-4x+1\right)-2bx+b-c=6x-6\)
\(\Leftrightarrow2bx-8+4ax-a+b+2=6x-6\)(Do c=-2)
\(\Leftrightarrow x\left(2b+4a\right)-\left(a-b+6\right)=6x-6\)
Do thỏa mãn với mọi x nên:
\(\left\{{}\begin{matrix}2b+4a=6\\a-b+6=6\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}b+2a=3\\a-b=0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}a=1\\b=1\end{matrix}\right.\)
Ta có :
\(\left\{{}\begin{matrix}a=1\\b=1\\c=-2\end{matrix}\right.\Leftrightarrow a+b+c=0\)
=> \(P\left(x\right)=x^2+x-2\)
