Eliminating \(y\) gives \(2x^2+(2m-6)x+m-4=0\). Its discriminant is \[\Delta=(2m-6)^2-8(m-4)=4(m^2-8m+17)=4((m-4)^2+1)>0\] for every real \(m\). The quadratic therefore has two distinct real roots, so the line has two distinct intersections with the curve.