let y = p(x) be the quartic and y = q(x) be the quintic polynomial function

finding the number of solutions to p(x) = q(x) will find the number of intersections of the two graphs

this is also equivalent to the number of roots of p(x) - q(x) = 0

since q(x) are is fifth degree polynomial and p(x) is fourth degree polynomial then p(x) - q(x) will be at most five degree polynomial

by the Fundamental Theorem of Algebra, a quintic polynomial can have at most 5 real solutions, thus the answer is 5.