Question 1183695
Without using calculus, a simpler solution would be as follows:


Equate {{{y = x^2 + 5x + 2}}} and {{{y = x + k}}}.

===> {{{x^2 + 5x + 2 = x + k}}}, since at their intersection point(s) they're supposed to have the same y-values.

<===> {{{x^2 + 4x + (2-k) = 0}}}.


Since the line is supposed to be tangent to the curve, it means that there are two identical roots for the preceding equation.  

This only happens when the discriminant is equal to 0, i.e.,


{{{4^2 - 4*1*(2-k) = 0}}}  ===> {{{4^2 = 4(2-k)}}} ===> {{{highlight(k = -2)}}}.


Solved.