Solve, `(1-sqrt(21-4x-x^(2)))/(x+1) ge 0`.

Answer

`x in [-2-2sqrt(6),-1) uu [-2+2sqrt(6),3]`

Explanation

We have `(1-sqrt(21-4x-x^(2)))/(x+1)ge0`
We must have `21-4x-x^(2) ge 0`
`implies x^(2)+4x-21 le 0`
`implies (x+7)(x-3) le 0`
`implies x in [-7,3]`
Case I : `x+1 gt 0 implies x gt -1` .......`(i)`
So given inequality reduces to
`1 ge sqrt(21-4x-x^(2))`
`implies 1 ge 21-4x-x^(2)`
`implies x^(2)+4x-20 ge 0`
`implies (x+2)^(2)-(2sqrt(6))^(2) ge 0`
`implies (x+2+2sqrt(6))(x+2-2sqrt(6)) ge 0`
`implies x in [-2+2sqrt(6),3]` ............`(ii)`
Case II : `x lt -1`
So given inequality reduces to
`1-sqrt(21-4x-x^(2)) le 0`
`implies 1 le sqrt(21-4x-x^(2))`
`implies 1 le 21-4x-x^(2)`
`implies x^(2)+4x-20 le 0`
`implies x in [-2-2sqrt(6),-1)` ...........`(iii)`
So from `(ii)` and `(iii)`,
`x in [-2-2sqrt(6),-1)uu[-2+2sqrt(6),3]`