The sum of the $y$-intercepts of the tangents drawn from the point $\left(-2,-1\right)$ to the hyperbola $\frac{x^2}{3}-\frac{y^2}{2}=1$ is

Let, the equation of the tangent from $\left(-2,-1\right)$ is $y=mx+\sqrt{3m^2-2}$
$\Rightarrow -1=-2m+\sqrt{3m^2-2}$
$\Rightarrow {\left(2m-1\right)}^2=3m^2-2$
$\Rightarrow m^2-4m+3=0$
$\Rightarrow m=1,3$
So, the equations of the tangents are
$\left(y+1\right)=\left(x+2\right) \& \left(y+1\right)=3\left(x+2\right)$
$\Rightarrow y$-intercepts of the tangents are $1$ & $5$
$\Rightarrow$ the sum of $y$-intercepts $=6$