Is this polynomial function quintic or cubic?

This only has 3 zeroes at the most. It is of degree three. It would ONLY be degree 5 if there was an x^ 5 in your equation.

I'll show you it only has 3 at -most- zeroes.
-----------------pull one x out--pull the second out.
x^3 + x^2 = (x)(x^2+x) =  (x)(x)(x+1)

Notice that two of the zeroes are exactly the same, however it is still of degree 3 since it has 3 factors. It technically only has 2 zeroes because 2 of them are equal.

A functions degree is the highest number exponent. Adding the exponents up gives you no useful information and will never help you.