Ashton designs movie sets. for one scene, he has to hang a banner across an archway that takes the shape of a parabola modeled by the quadratic equation y = −x2 + 6x. the banner is to be hung at an angle defined by the linear equation y − x = 4, as shown in the figure.

No question is asked so I'm not sure what you are looking for but below I calculated the point where the two equations intersect:
y - x = 4 → y = x + 4
y = - x² + 6x
x + 4 = - x² + 6x
x² - 5x + 4 = 0
(x - 4)(x - 1) = 0
 x = 4, x = 1

when x = 4, then y = x + 4 = 4 + 4 = 8 → (4,8)
when x = 1, then y = x + 4 = 1 + 4 = 5 → (1,5)

The line and parabola intersect at two points: (4,8) and (1,5)

Ineed show the photo because i need more information