The general tangent equation for $x^2+y^2+2gx+2fy+c=0$ at $(x_1, y_1)$ is $xx_1+yy_1+g(x+x_1)+f(y+y_1)+c=0$. For the circle $x^2+y^2-4x+6y-12=0$, we have $g=-2, f=3, c=-12$. At point $(5,1)$: $x(5)+y(1)+(-2)(x+5)+3(y+1)+(-12)=0$. This simplifies to $5x+y-2x-10+3y+3-12=0$. Combining terms gives $3x+4y-19=0$. This is the equation of the tangent.