The two circles $x^2+y^2-2x-3=0$ and $x^2+y^2-4x-6y-8=0$ are such that

The coordinates of centres $C_1$ and $C_2$ of two circles are $(1, 0)$ and $(2, 3)$ respective. Let $r_1$ and $r_2$ be the radii of two circles. Then, $r_1=2$ and $r_2=\sqrt{21}.$

Clearly $r_1-r_2<C_1{C}_2<r_1+r_2$

Hence, the two circles intersect each other.