At what value of p do they intercept?

Question

-2x^2 + 4x + 1 and x^2 + px + 4  

Answer 1

I don't see a simple way of answering this question except by graphing which will get you an answer quickly.

-2x²+4x+1 is a parabola with maximum point at x=1.

x² + px + 4 is a parabola with the minimum point at x = -p/2

If you take p = -2, you will see that the 2 graphs touch at x=1

Also if you explore the graphs with a graphing calculator which allows "sliders" (i.e. using parameters in the graph), you will see that there are many values of p which cause the graphs to intersect in one or more points.