2x4−x3+3x2−x+4x2−3x+2=f(x)+Ax−1+Bx−2 then

2x4−x3+3x2−x+4(x−1)(x−2)=2x2+5x+14+Ax−1+Bx−2Consider 31x−24(x−1)(x−2)=Ax−1+Bx−2 ⇒A=−7,B=38 ∴A+B=31

2x4−x3+3x2−x+4x2−3x+2=f(x)+Ax−1+Bx−2 then

2x4−x3+3x2−x+4(x−1)(x−2)=2x2+5x+14+Ax−1+Bx−2Consider 31x−24(x−1)(x−2)=Ax−1+Bx−2 ⇒A=−7,B=38 ∴A+B=31

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$\frac{2 x^{4} - x^{3} + 3 x^{2} - x + 4}{x^{2} - 3 x + 2} = f (x) + \frac{A}{x - 1} + \frac{B}{x - 2}$ then

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$f (x) = 2 x^{2} + 5 x + 14, A + B = 39$

$f (x) = 2 x^{2} - 5 x + 14, A + B = 31$

$f (x) = 2 x^{2} + 5 x + 14, A + B = 31$

$f (x) = 2 x^{2} + 5 x + 14, A = 4, B = 35$

answer is C.

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$\frac{2 x^{4} - x^{3} + 3 x^{2} - x + 4}{(x - 1) (x - 2)} = 2 x^{2} + 5 x + 14 + \frac{A}{x - 1} + \frac{B}{x - 2}$
Consider $\frac{31 x - 24}{(x - 1) (x - 2)} = \frac{A}{x - 1} + \frac{B}{x - 2}$
$\Rightarrow A = - 7, B = 38 ∴ A + B = 31$

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2x4−x3+3x2−x+4x2−3x+2=f(x)+Ax−1+Bx−2 then

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$\frac{2 x^{4} - x^{3} + 3 x^{2} - x + 4}{x^{2} - 3 x + 2} = f (x) + \frac{A}{x - 1} + \frac{B}{x - 2}$ then