x2+5x+6x2+x+2<0⇒(x+2)(x+3)1<0
$\begin{aligned}
& x \in(-3,-2) \ldots \ldots \ldots . .(1) \
& f(x)=1+x\left(\lambda^2-x^2\right)
\end{aligned}$
Finding local minima f′(x)=(λ2−x2)+(−2x)⋅x Put f′(x)=0 $\begin{aligned}
& \Rightarrow \lambda^2=3 x^2 \
& \Rightarrow x= \pm \frac{\lambda}{\sqrt{3}}
\end{aligned} We want local min\Rightarrow \mathrm{x}=\frac{-\lambda}{\sqrt{3}}from(1)x \in(-3,-2)\begin{aligned} & -3 < \frac{-\lambda}{\sqrt{3}} < -2 \ & 3 \sqrt{3}>\lambda>2 \sqrt{3} \ & \alpha=2 \sqrt{3}, \beta=3 \sqrt{3} \ & \alpha^2+\beta^2=12+27=39\end{aligned}$