Find all values of the parameter a for which the quadratic equation
`(a+1)x^(2)+2(a+1)x+a-2=0`
(i) has two distinct roots.
(ii) has no roots.
(iii) has to equal roots.

Explanation

By the hypothesis, this equation is quadratic and therefore `a!=-1` and the discriminant of this equation
`D=4(a+1)^(2)-4(a+1)(a-2)`
`=4(a+1)(a+1-a+2)`
`=12(a+1)`
(i) For `agt(-1)` then `Dgt0`, this equation has two distinct roots.
(ii) For `a lt(-1)`, then `Dlt0`, this equation has no roots.
(iii) This equation cannot have two equal roots. Sicne, `D=0` only for `a=-1` and this contradicts the hypothesis.