If (2x-3)(x^2-4x+1)=Ax^3+Bx^2+(x+1) find the value of A+B+C+D

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The problem provides inconsistent information (there are not given C and D), hence, supposing that C is the coefficient of (x+1) and D is the constant term yields:

`2x*x^2 + 2x*(-4x) + 2x*1 - 3*x^2 - 3*(-4x) - 3*1 = Ax^3+Bx^2+C(x+1) + D`

`2x^3 - 8x^2 + 2x - 3x^2 + 12x...

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The problem provides inconsistent information (there are not given C and D), hence, supposing that C is the coefficient of (x+1) and D is the constant term yields:

`2x*x^2 + 2x*(-4x) + 2x*1 - 3*x^2 - 3*(-4x) - 3*1 = Ax^3+Bx^2+C(x+1) + D`

`2x^3 - 8x^2 + 2x - 3x^2 + 12x - 3 = Ax^3+Bx^2+C(x+1) + D`

You need to collect like terms such that:

`2x^3 - 11x^2 + 14x - 3 = Ax^3+Bx^2+C(x+1) + D`

Equating the coefficients of like powers yields:

`A = 2 ; B = -11 , C = 14`

`C + D = -3 =gt 14 + D = -3 =gt D = -17`

Hence, evaluating A,B,C,D under given conditions yields `A = 2 ; B = -11 , C = 14, D = -17.`

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