We have the given equation
$\lambda x^{2}+4 x y +y^{2}+\lambda x+3 y+2=0$
On comparing this equation with
$a x^{2}+2 h x y +b y^{2}+2 g x+2 f y +c=0$, we get
$a=\lambda, h=2,\, b=1,\, g=\frac{\lambda}{2},\, f=\frac{3}{2}, c=2$
Since, given equation represents a parabola
$\therefore h^{2} =a b$
$\Rightarrow 4=\lambda \cdot 1$
$\Rightarrow \lambda=4$