The center of circle #(x-4)^2+(y+3)^2=9# is #(2,-3)# and radius is #3# and center of circle #(x+4)^2+(y-1)^2=16# is #(-4,1)# and radius is #4#.

The distance between centers is #sqrt((-4-2)^2+(1-(-3))^2#

= #sqrt(36+16)=sqrt52=7.2111#

If the radii of two circles is #r_1# and #r_2# and we also assume that #r_1>r_2# and the distance between centers is #d#, then

**A** - if #r_1+r_2=d#, they touch each other externally and greatest possible distance between a point on one circle and another point on the other is #2(r_1+r_2)# and smallest possible distance between a point on one circle and another point on the other is #0#.

**B** - if #r_1+r_2 < d#, they do not touch each other and are outside each other (i.e. one is not contained in other) and greatest possible distance between a point on one circle and another point on the other is #r_1+r_2+d# and smallest possible distance between a point on one circle and another point on the other is #d-r_1-r_2#.

**C** - if #r_1+r_2 > d# and #r_1-r_2=d#, they touch each other internally and smaller circle is contained in other and greatest possible distance between a point on one circle and another point on the other is #2r_2# and smallest distance is #0#.

**D** - if #r_1+r_2 > d# and #r_1-r_2>d#, smaller circle lies inside the larger circle and greatest possible distance between a point on one circle and another point on the other is #r_1+r_2+d# and smallest distance is #r_1-r_2-d#.

**E** - if #r_1+r_2 > d# and #r_1-r_2 < d#, the two circles intersect each other and greatest possible distance between a point on one circle and another point on the other is #r_1+r_2+d# and smallest distance is #0#.

Now, here as #r_1+r_2=7 < d=7.2111# and #r_1-r_2 < d#, they do not touch each other and are outside each other (i.e. one is not contained in other) and greatest possible distance between a point on one circle and another point on the other is #4+3+7.2111=14.2111#.

graph{(x^2+y^2-8x+6y+16)(x^2+y^2+8x-2y+1)=0 [-10.5, 9.5, -5.665, 4.755]}