# The sum of values of x satisfying the equation |x^2+4x+3|+2x+5=0 is

Solution:    If  $x^{2}+4x+3=(x+3)(x+1)\geq 0,x \in R -(-3,-1)$     $......(1)$

The given equation  becomes     $x^{2}+6x+8=0$

$\Rightarrow$                      $x=-2,-4$             $.........(2)$$.........(2)$

From  $(1),(2)$       $\Rightarrow$                $x=-4$

If    $x^{2}+4x+3< 0,x\in (-3,-1)$          $........(3)$

The equation becomes       $-(x^{2}+4x+3)+2x+5=0$

$x^{2}+2x-2=0\Rightarrow x=-1\pm \sqrt{3}$       $.........(4)$

From $(3),(4)$       $\Rightarrow$           $x=-1-\sqrt{3}$

Sum of the roots $=-4+(-1-\sqrt{3})=-5-\sqrt{3}.$

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