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$Q10$  The equation $p\left ( x \right ) = \alpha$ where $p\left ( x \right ) = x^{4}+4x^{3}-2x^{2}-12x$ has four distinct real root if and only if

  1. $p\left ( -3 \right )<\alpha$
  2. $p\left ( -1 \right )>\alpha$
  3. $p\left ( -1 \right )<\alpha$
  4. $p\left ( -3 \right )<\alpha <p\left ( -1 \right )$
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Alternate form of equation is$:$

$x(x+2)(x^{2}+2x-6)=0$ $\text{ or }$ $x(x^{3}+4x^{2}-2x-12)=0$

As $p(x) = \alpha$

Roots of the equation  $\text{or}$  value of $\alpha$ are$:$

$x = - 2, x = 0, x = - 1 - \sqrt{7}, x = \sqrt{7} -1$

$p(-3) = - 9$

$p(-1) = 7$

Hence option $\text{D}$ is correct.
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