15. For which $m$, the roots of the equation $x^{2}-2 m x+m^{2}-1=0$ are in the interval ( $-2,4)$ ?
Solution. The roots of the equation

$$
x^{2}-2 m x+m^{2}-1=0
$$

are

$$
x_{1,2}=\frac{2 m \pm \sqrt{4 m^{2}-4 m^{2}+4}}{2}=m \pm 1 \text { i.e. } x_{1}=m-1 \text { and } x_{2}=m+1 \text {. }
$$

Since the roots must be in the interval $(-2,4)$, we have

$$
\left\{\begin{array}{l}
-2<m-1<4 \\
-2<m+1<4
\end{array}\right.
$$

and from this

$$
\left\{\begin{array}{c}
-1<m<5 \\
-3<m+1<3
\end{array}\right.
$$

i.e. $-1<m<3$.