Interesting!

Alternatively, we can also transform all the equations themselves in natural numbers, since each equations is of the form

$ \pm a_nx^n\pm ...\pm a_2x^2 \pm a_1x\pm a_0=0 $

representing a set of constructible numbers, aka algebraic numbers, first lay down the exponents like this

$ \pm a_nxn\pm ... \pm a_2x2\pm a_1x\pm a_0=0 $

second, let $ x, +, -, = $ be new symbols representing each a numeral, them each equation turns into a numeral in the $ base_{14} $, so that

$ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, x, +, -, = $

are the numerals in $ base_{14} $, now trivially any numerical base is denumerable with $ base_{10} $.

Example:

$ 3x^4-7x^2+1=0 $ is $3x4-7x2+1=0$ in $ base_{14} $ and so 1081600743426 in $ base_{10} $.

The algebraic numbers are countable. You can create an index $i$ for the roots of a polynomial $$p(x) = \sum_{k=0}^n a_kx^k \qquad (a_k \in \mathrm Z)$$ by $$i = n + \sum_{k=0}^n |a_n|$$

There are then countably many indexes (no more than the natural numbers) and finitely many roots having each index. Since and countable collection of countable sets is itself countable, the job is done.