If the quadratic equation \((\lambda +2){\mathrm{x}}^{2}-3\mathrm{λx}+4\lambda =0,\lambda \neq -2\), has two positive ro…

\( f(x)=(\lambda+2) x^2-3 \lambda x+4 \lambda\)

Case - 1: \(\frac{c}{a}>0\) or \(a c>0\)

\( (\lambda+2) 4 \lambda>0\)

\( \Rightarrow \lambda<-2 \text { or } \lambda>0\)

Case - 2: \(-\frac{\mathrm{b}}{2 \mathrm{a}}>0 \)
\( \frac{3 \lambda}{2(\lambda+2)}>0 \)
\( \Rightarrow \lambda<-2 \text { or } \lambda>0\)
Case - 3: \(\mathrm{D} \geq 0\)
\( (-3 \lambda)^2-4(\lambda+2) \times 4 \lambda \geq 0 \)
\( \lambda(7 \lambda+32) \leq 0 \)
\( \Rightarrow \lambda \in\left[\frac{-32}{7}, 0\right]\)
Intersection of Case-1, Case-2 and Case-3

\( \Rightarrow \lambda \in\left[\frac{-32}{7},-2\right)\)
\( \Rightarrow \lambda \in[-4.57,-2)\)
\( \Rightarrow \lambda=-4,-3\)
Number of values of \(\lambda=2\)