I'm trying to solve a simple inequality, that will show from what n is an exponential function greater than a polynomial one.
For example consider the following functions
$f(n) = 3^n$
$g(n) = 6n^3 + 6n^2 + 20n + 16$
It's pretty clear that $f(n)$ is growing faster than $g(n)$, and it's pretty easy to brute-force and find a natural number from which $f(n) > g(n)$.
However - I wonder if there's a general case to solve these kind of inequalities since I've never done this before.
Generally exponentials do not play nicely with polynomials, so you are forced to do a numerical solution. Ignoring the small terms, you should be able to solve $3^n=6n^3$ using the Lambert W function, but I haven't studied it and when I see solutions with it I don't feel any smarter. In this case $n$ is probably not very large, so the small terms may not be so small.
As long as the base of the exponential is not very close to $1$, the power will grow very quickly so just checking is quick. Here we already have $3^7=2187 \gt 6\cdot 7^3=2058$. The smaller terms make the polynomial larger here, but by $8$ the exponential is much larger and never looks back.