Question

if p(x)= (x-1)(x+k) and if the line tangent to the graph of p at the point (4, p(4)) is parallel to the line 5x-y+6=0, then k=?

Answer 1

Notice that the line 5x-y+6=0 has slope 5. Hence the tangent line at (4,p(4)) has slope 5 as well. It means that p'(4)=5.
But on the other hand p'(x)=(x+k)+(x-1)=2x+k-1. So
p'(4)= 7+k=5.
Therefore k=-2.