Which property of polynomial addition says that the sum of two polynomials is always a polynomial

The property of polynomial addtion that says that the sum of two polinomials is alwasy a polynomial is called closure property of addition or under addtition.

Polynomials are closed under addtiion because when you add polynomials the letters and their exponents do no change, you just add the coefficients of the like terms (those with same letters raised to the same exponents), so the result will be other polinomyal of the same kind, except for the terms that cancel (positve with negative) which does not change the fact that the result is still a poynomial.

In mathematics the closure property means that the result of an operation over a kind of "number" will result in a "number" of the same kind.