The equation of the circle has the form (x+3)²+(y-1)²=a² where a is the radius.
The circle must pass through (-2,5), so 1²+4²=17=a².
The equation is (x+3)²+(y-1)²=17.
Differentiating we get 2(x+3)+2(y-1)dy/dx=0, dy/dx=-(x+3)/(y-1).
The tangent is dy/dx so at (-2,5), dy/dx=-1/4.
Tangent line is y-5=-(x+2)/4=-x/4-1/2, y=-x/4+9/2.