Is it possible when multiplying two polynomials that, after collecting similar terms, all terms vanish?

If you are working with polynomials over a ring with zero divisors, such as $\mathbb{Z}/4\mathbb{Z}$, then it is possible for the product of two polynomials to vanish. This may be what Gelfand is coyly alluding to. But in the ordinary sense of polynomials with rational, real, or complex coefficients, the degree of the product is the sum of the degrees of the polynomials being multiplied together.

Just look at the highest-order coefficient $-$ it can't be zero, because it is the product of the highest-order coefficients of $p$ and $q$.

Consider the zero polynomial $p(x)=0$, then certainly multiplying any real polynomial by $p(x)$ yields $0$.

For a polynomial with $\deg\geq0$, over some field $\mathbb{F}$, this is not possible.

Consider the general polynomial of the form:

$$p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_o$$

For $a_n$ having real or complex coefficients.

Clearly, as you multiply two polynomials, the $\deg$ of each term increases, except for the constant case, which yields a proportional polynomial.

So we can simply say, for all $x\in \mathbb{R}$, such that two real polynomials $p(x)g(x)=0$ implies $p(x) \ or \ g(x)=0$.