Question 261485
Because the curve {{{y=(kx+8)/(k+x)}}} has a tangent line of {{{y=x+4}}} at {{{x=-2}}}, this means that the curve intersects the line at {{{x=-2}}} and they intersect at the same 'y' value. In other words, the curve and the line intersect at the point (-2, y). Because these y values are the same, we can plug {{{y=x+4}}} into {{{y=(kx+8)/(k+x)}}} to get {{{x+4=(kx+8)/(k+x)}}}. 



Now plug in {{{x=-2}}} (since they intersect at x=-2) to get {{{-2+4=(-2k+8)/(k-2)}}} and simplify to get {{{2=(-2k+8)/(k-2)}}}. From here, it's just a simple matter of solving for k which I'll let you do.