Roots in an Interval

1. Factorize the Quadratic Equation

The given quadratic equation is:

$$x^2 - 2mx + m^2 - 1 = 0$$

Notice that the first three terms form a perfect square:

$$(x - m)^2 - 1 = 0$$

2. Find the Roots Directly

Isolate the perfect square term:

$$(x - m)^2 = 1$$

Take the square root of both sides:

$$x - m = \pm 1$$

This gives the two distinct roots directly:

$$\alpha = m - 1$$

$$\beta = m + 1$$

3. Apply the Given Boundary Conditions

The problem states that both roots must be strictly greater than $$-2$$ and strictly less than $$4$$.

For the smaller root $$\alpha = m - 1$$ to be greater than $$-2$$:

$$m - 1 > -2$$

$$m > -1$$

For the larger root $$\beta = m + 1$$ to be less than $$4$$:

$$m + 1 < 4$$

$$m < 3$$

4. Combine the Inequalities

Intersecting the two boundary requirements gives the final range for the parameter:

$$-1 < m < 3$$

Final Answer

All the values of $$m$$ lie in the interval $$(-1, 3)$$.