The equation of the tangent at $\left(–\mathrm{3,2}\right)$ to the parabola $y^2+4x+4y=0$ is 
$2y+2\left(x-3\right)+2\left(y+2\right)=0$
or $2\mathrm{x}+4\mathrm{y}-2=0\Rightarrow \mathrm{x}+2\mathrm{y}-1=0$
Since the tangent at one end of the focal chord is parallel to the normal at the other end, the slope of the normal at the other end of the focal chord is $-\frac{1}{2}$