for what values of m, the equation ` 2x^(2) -2 ( 2m +1) x+ m ( m +1) = 0 m in R` has (i) Both roots smallar than 2 ? (ii) Both roots greater than 2 ? (iii) Both roots lie in the interval (2,3) ? (iv) Exactly one root lie in the interval (2,3) ? (v) One root is smaller than 1, and the other root is greater than 1 ? (vi) One root is greater than 3 and the other root is smaller than 2 ? (vii) Roots ` alpha and beta` are such that both 2 and 3 lie between `alpha and beta ` ?

Explanation

To solve the given quadratic equation \( 2x^2 - 2(2m + 1)x + m(m + 1) = 0 \) for the various conditions specified, we will analyze the roots based on the properties of quadratic equations.

### Step-by-Step Solution

1. **Identify the coefficients**:
The given quadratic equation can be rewritten in the standard form \( ax^2 + bx + c = 0 \) where:
- \( a = 2 \)
- \( b = -2(2m + 1) = -4m - 2 \)
...