The number of integral roots of the equation $x (x + 1) (x + 2) (x + 3) = 120$ is

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Solution

$x (x + 1) (x + 2) (x + 3) = 120 \Rightarrow (x + 1) (x + 2) x (x + 3) = 120 \Rightarrow (x^{2} + 3 x + 2) (x^{2} + 3 x) = 120$

Let $x^{2} + 3 x = y$
Then, $(y + 2) y = 120$
$\Rightarrow y^{2} + 2 y - 120 = 0 \Rightarrow (y + 12) (y - 10) = 0 \Rightarrow y = - 12, 10$

When $y = - 12,$
$x^{2} + 3 x = - 12 \Rightarrow x^{2} + 3 x + 12 = 0 D = 9 - 48 < 0$
No real roots

When $y = 10,$
$x^{2} + 3 x = 10 \Rightarrow x^{2} + 3 x - 10 = 0 \Rightarrow (x + 5) (x - 2) = 0 ∴ x = - 5, 2$

Hence, the number of integral solutions is $2$ .

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