The correct option is D 3x−1x2+x+1+2(x+1)2−3(x+1)
Let x3−3x−2(x2+x+1)(x+1)2=A(x+1)+B(x+1)2+Cx+D(x2+x+1)
⇒x3−3x−2=A(x+1)(x2+x+1)+B(x2+x+1)+(Cx+D)(x+1)2⇒x3−3x−2=A(x3+2x2+2x+1)+B(x2+x+1)+C(x3+2x2+x)+D(x2+2x+1)
On comparing coefficients
A+C=1,2A+B+2C+D=0,2A+B+C+2D=−3,A+B+D=−2⇒A=−3,B=2,C=3,D=−1
Hence
x3−3x−2(x2+x+1)(x+1)2=−3(x+1)+2(x+1)2+3x−1(x2+x+1)
Hence, option 'C' is correct.