jmc

Multiplying both sides by $x-4,$ we get $x(x-4)+45=-10(x-4),$ or $x^2-4x+45=-10x+40,$ which simplifies to $x^2+6x+5=0.$ This quadratic factors as $(x+1)(x+5)=0,$ so either $x=-1$ or $x=-5,$ both of which we can check are valid solutions. Therefore, the answer is $$x=\boxed{-1,-5}.$$