The partial fractions of x3−5x2−3x+2 are

x + 3 + 4 x − 1 − 3 x − 2

x + 3 + 4 x − 1 − 3 x − 2

x + 3 − 4 x − 1 + 3 x − 2

x + 3 − 4 x − 1 − 2 x − 2

x + 3 + 4 x − 1 − 3 x − 2

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The partial fractions of $\frac{x^{3} - 5}{x^{2} - 3 x + 2}$ are

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x + 3 - \frac{4}{x - 1} + \frac{3}{x - 2}

x + 3 - \frac{4}{x - 1} - \frac{2}{x - 2}

x + 3 + \frac{4}{x - 1} + \frac{3}{x - 2}

x + 3 + \frac{4}{x - 1} - \frac{3}{x - 2}

answer is D.

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\frac{x^{3} - 5}{x^{2} - 3 x + 2} = (x + 3) + \frac{7 x - 11}{x^{2} - 3 x + 2}

$\frac{7 x - 11}{x^{2} - 3 x + 2} = \frac{7 x - 11}{(x - 1) (x - 2)} = \frac{A}{x - 1} + \frac{B}{x - 2}$

$7 x - 11 = A (x - 2) + B (x - 1)$

Put $x = 1$

$- 4 = - A \Rightarrow A = 4$

Put $x = 2 \Rightarrow B = 3$

$∴ \frac{x^{3} - 5}{x^{2} - 3 x + 2} = (x + 3) + \frac{4}{x - 1} + \frac{3}{x - 2}$

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