Question

Solve $|x^2+4x+3|+2x+5=0$

Answer 1

Given, $\left|x^2+4x+3\right|+2x+5=0$
Case $\mid x^2+4x+3>0\Rightarrow (x<-3$ or $x>-1)$
$\therefore x^2+4x+3+2x+5=0$
$\Rightarrow x^2+6x+8=0$
$\Rightarrow (x+4)(x+2)=0$
$\Rightarrow x=-4,-2[\text{ but }x<-3\text{ or }x>-1]$
$\therefore x=-4$ is the only solution. ...(i)
Case II $x^2+4x+3<0\Rightarrow (-3<x<-1)$
$\therefore -x^2-4x-3+2x+5=0$
$\Rightarrow x^2+2x-2=0\Rightarrow (x+1{)}^2=3$
$\Rightarrow |x+1|=\sqrt{3}$
$\Rightarrow x=-1-\sqrt{3},-1+\sqrt{3}[\text{ but }x\in (-3,-1)]$
$\therefore x=-1-\sqrt{3}$ is the only solution . ..(ii)
From Eqs. (i) and (ii), we get
$x=-4$ and $(-1-\sqrt{3})$ are the only solutions.