Example 2. Let's determine in which numeral system the following multiplication was performed: $352 \cdot 31=20152$.
Solution. Let $x$ be the base of the numeral system, then the given equality can be written in the form of an equation

$$
\left(3 x^{2}+5 x+2\right)(3 x+1)=2 x^{4}+x^{2}+5 x+2
$$

By performing the multiplication and combining like terms, we get:

$$
2 x^{4}-9 x^{3}-17 x^{2}-6 x=0
$$

It is clear that $x \neq 0$ and therefore the equation will take the form:

$$
2 x^{3}-9 x^{2}-17 x-6=0
$$

Since $x$ as the base of the numeral system can only take natural values, among the divisors of the constant term, only the numbers $1,2,3,6$ need to be tested. If we also consider that the digit 5 is the largest in the given equality, then only $x=6$ needs to be checked.

Let's divide the polynomial $2 x^{3}-9 x^{2}-17 x-6$ by $(x-6)$ using Horner's scheme:

6 | 2 | -9 | -17 | -6 |
| ---: | ---: | ---: | ---: |
| 2 | 3 | 1 | 0 |

We obtained zero as the remainder. This means that $x=6$ is a root of the obtained equation and is the base of the desired numeral system.