9.103. For what values of $m$ are the roots of the equation $4 x^{2}-(3 m+1) x-m-2=0$ contained in the interval between -1 and 2?
## Solution.

The roots of the equation $a x^{2}+b x+x=0$ are contained in the interval $(\lambda, \delta)$, i.e. $\lambda 0 , } \\
{ a ( a \lambda ^ { 2 } + b \lambda + c ) > 0 , } \\
{ \lambda 0, \\
4(-1)^{2}-(3 m+1)(-1)-m-2>0, \\
-1 0 , } \\
{ 2 m + 3 > 0 , } \\
{ 3 m - 1 5  0 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
m \in R \\
m > -\frac{3}{2} \\
m > -3
\end{array} \Leftrightarrow\left(-\frac{3}{2}<m<\frac{12}{7}\right)\right.\right.
$$

Answer: $m \in\left(-\frac{3}{2} ; \frac{12}{7}\right)$.