The factors of $x^{3} - 2 x^{2} - 13 x - 10$ are

(x - 1) (x + 2) (x + 5)

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(x - 1) (x - 2) (x - 5)

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(x + 1) (x - 2) (x + 5)

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(x + 1) (x + 2) (x - 5)

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Solution

The correct option is D $(x + 1) (x + 2) (x - 5)$
Let $p (x) = x^{3} - 2 x^{2} - 13 x - 10$
By trail, we find that $p (- 1) = (- 1)^{3} - 2 (- 1)^{2} - 13 (- 1) - 10 = - 1 - 2 + 13 - 10 = 0$ . So $(x + 1)$ is factor of $p (x)$ .
Now, $x^{3} - 2 x^{2} - 13 x - 10 = x^{3} + (x^{2} - 3 x^{2}) - (3 x + 10 x) - 10$
$= (x^{3} + x^{2}) - (3 x^{2} + 3 x) - (10 x + 10)$
$= x^{2} (x + 1) - 3 x (x + 1) - 10 (x + 1)$
$= (x + 1) (x^{2} - 3 x - 10)$
$= (x + 1) [x^{2} - 5 x + 2 x - 10]$
$= (x + 1) [x (x - 5) + 2 (x - 5)]$
$= (x + 1) (x - 5) (x + 2)$
$= (x - 5) (x + 1) (x + 2)$
Option D is correct.

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