# Core 1 algebra question that's completely bamboozled me...

This $kx-4$ can only be a tangent when the equation $x^2+x=kx-4$ has exactly one solution, that is, like you stated when the discriminant of $f(x)=x^2+(1-k)x+4=0$ is equal to zero.
Hence you must find k such that $\Delta(f) = (1-k)^2-16= 0$