Solve `|x^2+4x+3|+2x+5=0.`

Answer

`-4and (-1-sqrt3)`

Explanation

Given, `|x^(2)+4x+3|+2x+5=0`
Case I `x^(2)+2+4x+3|gt0implies(xlt-3or x gt-1)`
`thereforex^(2)+4x+3+2x+5=0`
`impliesx^(2)+6x+8=0implies(x+4)(x+2)=0`
`impliesx=-4,-2" "[but x lt-3 or x gt-1]`
`thereforex=4" is not only solution."" "...(i)`
Case II `x^(2)+4x+3lt0implies(-3ltxlt-1)`
`therefore-x^(2)-4x-3+2x+5=0`
`impliesx^(2)+2x-2=0implies(x+1)^(2)=3`
`implies|x+1|=sqrt3`
`impliesx=-1-sqrt3,-1+sqrt3" "[butx in(-3,-1)]`
`thereforex=-1-sqrt3 "is the only solution." " "...(ii)`
From Eqs. (i) and (ii), we get
`x=-4 and (-1-sqrt3)` are the only solutions.