Explanation

(c)
$(x + 2)(x + 12)(x + 3)(x + 8) = 4x^{2}\$.
$\Rightarrow \left( x^{2} + 14x + 24 \right)\left( x^{2} + 11x + 24 \right) = 4x^{2}$
$\Rightarrow \left( \ x + 14 + \frac{24}{x} \right)\left( \ x + 11 + \frac{24}{x} \right) = 4$
Put $x + \frac{24}{x} = y$
$(y + 14)(y + 11) = 4$
$\Rightarrow y^{2} + 25y + 154 - 4 = 0$
$\Rightarrow y^{2} + 25y + 150 = 0$
$\Rightarrow y^{2} + 15y + 10y + 150 = 0$
$\Rightarrow y(y + 15) + 10(y + 15) = 0$
$\Rightarrow y = - 10,\ - 15$
$\Rightarrow x + \frac{24}{x} = - 10,\ x + \frac{24}{x} = - 15$
$\Rightarrow x^{2} + 10x + 24 = 0,\ x^{2} + 15x + 24 = 0$
$\Rightarrow x^{2} + 6x + 4x + 24 = 0$
$\Rightarrow x(x + 6) + 4(x + 6) = 0$
$\Rightarrow x = - 4,\ - 6$
and $x^{2} + 15x + 24 = 0$
$\Rightarrow x = \frac{- 15 \pm \sqrt{225 - 96}}{2}$
$= \frac{- 15 \pm \sqrt{129}}{2}$
Number of integer root is 2.