No. Even if the answer is zero, zero is still a polynomial.
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You simply need to multiply EACH term in one polynomial by EACH term in the other polynomial, and add everything together.
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The "degree" is only specified for polynomials. The degree of a monomial (a single term) is the sum of the powers of all the variables. For example, x3y2z would have the degree 6; you have to add 3 + 2 + 1 (since z is the same as z to the power 1). The degree of a polynomial is the degree of its highest monomial.
To find the polynomial degree you have only to add the exponents of all of the different components of the polynomial. In your case, you would add 1 and 5 from 4ab5 to get 6, 1 and 1 from 2ab to get 2, and 4 and 3 from 3a4b3 to get 7. Since the degree of the third component is the highest, that is you're answer.
A polynomial is made up of one or several monomials (terms added or subtracted together). The term with the highest degree should have a degree of 4. To get the degree, if it's a single variable, the degree is the power to which it is raised; if there are several variables, add all the powers together.