\(P\left(1\right)=1+a+b+c=4\Rightarrow a+b+c=3\left(1\right)\)
\(P\left(-2\right)=-8+4a-2b+c=7\Rightarrow4a-2b+c=15\left(2\right)\)
\(\left(1\right);\left(2\right)\Rightarrow4a-2b+c-\left(a+b+c\right)=12\)
\(\Rightarrow3a-3b=12\)
\(\Rightarrow a-b=4\left(3\right)\)
\(M=P\left(4\right)-P\left(-5\right)+2051\)
\(\Rightarrow M=64+16a+4b+c-\left(-125+25a-5b+c\right)+2051=-9a+9b+2240\)
\(\Rightarrow M=-9\left(a-b\right)+2240=-9.4+2240=2204\)
Ta có: P(1)=4 và P(-2)=7
=>\(\left\{{}\begin{matrix}1^3+a\cdot1^2+b\cdot1+c=4\\\left(-2\right)^3+a\cdot\left(-2\right)^2+b\cdot\left(-2\right)+c=7\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}a+b+c=3\\4a-2b+c=7+8=15\end{matrix}\right.\)
=>\(4a-2b+c-a-b-c=15-3\)
=>3a-3b=12
=>a-b=4
=>b=a-4
=>\(P\left(x\right)=x^3+ax^2+\left(a-4\right)x+c\)
\(P\left(4\right)=4^3+a\cdot4^2+\left(a-4\right)\cdot4+c\)
\(=64+16a+4a-16+c\)
=20a+48+c
\(P\left(-5\right)=\left(-5\right)^3+a\cdot\left(-5\right)^2+\left(a-4\right)\cdot\left(-5\right)+c\)
=-125+25a-5a+20+c
=20a-105+c
M=P(4)-P(-5)+2051
=(20a+48+c)-(20a+c-105)+2051
=20a+48+c-20a-c+105+2051
=2051+105+48
=2204
