A rational function is a function that can be written as the quotient of two polynomials.
You did write:
"5x4−6x7+15x4−6x7+1"
This is not a rational function, but I guess that there is some kind of mistake in what you wrote, as the division sign may be missing.
A rational function would actually be:
Where this is considered a rational function because we have the quotient of two polynomials:
p(x) = 5*x^4 - 6*x^7
q(x) = 15*x^4 - 6*x + 1
If you want to learn more, you can read: