If x2+xx+1x−22x2+3x−13x3x−3x2+2x+32x−12x−1=xA+B then

Let Δ=x2+xx+1x−22x2+3x−13x3x−3x2+2x+32x−12x−1Applying R2→R2−2R1 and R3→R3−R1, we getΔ=x2+xx+1x−2x−1x−2x+1x+3x−2x+1=x2x+1x−20x−2x+10x−2x+1+xx+1x−2x−1x−2x+1x+3x−2x+1=0+xx+1x−2x−1x−2x+1x+3x−2x+1Applying R2→R2−R1 and R3→R3−R2, we get Δ=xx+1x−2−1−33400   =xxx−1−33400+01−2−1−33400=111−1−33400x+01−2−1−33400