Answer
Vertex: $(4,-3)$
Focus: $(3\frac{1}{2},-3)$
Directrix: $x=4\frac{1}{2}$
Work Step by Step
Recall: The parabola $(y-b)^2=4p(x-a)$ has a vertex $(a,b)$, a focus $(a+p,b)$, and a directrix $x=-p+a$.
We have $y^2+6y+2x+1=0$ or equivalently,
$y^2+6y+9+2x-8=0$
$(y+3)^2+2(x-4)=0$
$(y+3)^2=-2(x-4)$
$(y-(-3))^2=-2(x-4)$
Then,
$(a,b)=(4,-3)$
$4p=-2$
$p=-\frac{1}{2}$
So, the vertex is $(4,-3)$, the focus is $(3\frac{1}{2},-3)$, and the directrix $x=4\frac{1}{2}$.
Sketch the graph:
