When x=-2, y=5. This means the vertex of the parabola is at (-2,5), which is a maximum turning-point meaning that the parabola is an upturned U shape. On either side of the line x=-2 the curve looks the same, as if x=-2 were a mirror, so this is the axis of symmetry and the focus lies on this line.
The vertex lies midway between the focus and directrix, and we can call the distance of the vertex from the focus and directrix k, making the line y=5-k the directrix line and (-2,5+k) the focus. All parabolas obey the rule that a point P(x,y) on the curve is equidistant from the directrix line (perpendicular distance) and the focus.
The distance of P from the directrix is y-(5-k), and the distance from the focus is sqrt((x-(-2))^2+(y-(5+k))^2) (Pythagoras). These distances must be equal, so (y+k-5)^2=(x+2)^2+(y-k-5)^2. But (x+2)^2=-10(y-5), so (y+k-5)^2=-10(y-5)+(y-k-5)^2, -10(y-5)=(y+k-5+y-k-5)(y+k-5-(y-k-5)) (difference of two squares)=(2y-10)(2k). The factor y-5 drops out: -10=4k, so k=-10/4=-5/2. Therefore the directrix line is y=5+5/2=15/2 and the focus is (-2,5/2).