If the line y = 3x − p does not intersect the curve y = x2 + 4x, find the range of values of p.
They intersect where
x^2 + 4x = 3x - p
x^2 + x + p = 0
If there are no solutions, meaning no intersections, then the discriminant must be negative
1^2 - 4p < 0
1^2 < 4p
p > 1/4
You can see from the plot below that this is true
www.wolframalpha.com/input/?i=plot+y%3Dx%5E2%2B4x%2C+y%3D3x-1%2F4