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

To solve the quadratic equation \( (a+1)x^2 + 2(a+1)x + (a-2) = 0 \) for different conditions based on the parameter \( a \), we will analyze the discriminant and the nature of the roots.

### Step 1: Identify the coefficients
The given quadratic equation can be expressed in the standard form \( Ax^2 + Bx + C = 0 \), where:
- \( A = a + 1 \)
- \( B = 2(a + 1) = 2a + 2 \)
- \( C = a - 2 \)

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