2x4-x3+3x2-x+4x2-3x+2=fx+Ax-1+Bx-2 then

2x4-x2+3x2-x+4x2-3x+2 x2-3x+2)2x4-x3+3x2-x+4(2x2+5x+14 2x4-6x3+4x25x3-x2-x+4 5x3-15x2+10x14x2-10x+4 14x2-52x+2842x-24f(x)=2x2+5x+14 2x4-x3+3x2-x+4x2-3x+2=f(x)+Ax-1+Bx-2 2x4-x3+3x2-x+4=f(x) (x-1) (x-2)+A(x-2)+B(x-1) x=1 ⇒ 2-1+3-1+4=-A ⇒ A=-7, B=38 f(x)=2x2+5x+14, A+B=31

2x4-x3+3x2-x+4x2-3x+2=fx+Ax-1+Bx-2 then

2x4-x2+3x2-x+4x2-3x+2 x2-3x+2)2x4-x3+3x2-x+4(2x2+5x+14 2x4-6x3+4x25x3-x2-x+4 5x3-15x2+10x14x2-10x+4 14x2-52x+2842x-24f(x)=2x2+5x+14 2x4-x3+3x2-x+4x2-3x+2=f(x)+Ax-1+Bx-2 2x4-x3+3x2-x+4=f(x) (x-1) (x-2)+A(x-2)+B(x-1) x=1 ⇒ 2-1+3-1+4=-A ⇒ A=-7, B=38 f(x)=2x2+5x+14, 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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$f (x) = 2 x^{2} - 5 x + 14, A + B = 31$

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

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

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

answer is C.

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Detailed Solution

$\frac{2 x^{4} - x^{2} + 3 x^{2} - x + 4}{x^{2} - 3 x + 2} x^{2} - 3 x + 2) 2 x^{4} - x^{3} + 3 x^{2} - x + 4 (2 x^{2} + 5 x + 14 \frac{2 x^{4} - 6 x^{3} + 4 x^{2}}{5 x^{3} - x^{2} - x + 4} \frac{5 x^{3} - 15 x^{2} + 10 x}{14 x^{2} - 10 x + 4} \frac{14 x^{2} - 52 x + 28}{42 x - 24}$

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