I am studying for finals and in the review packet is shown this problem: $$P(x)=2x^4 + 5x^3 + 5x^2 + 20x - 12$$ I don't know what to do, I have already tried looking in the textbook and Khan academy.
The rational root test theorem says that, if rational factors of a polynomial exist, then they are always in the form of
$\pm$(factor of last coefficient) / (factor of first coefficient)
In this case, the factors you can try are: $\pm 12, \pm 6, \pm 4, \pm 3, \pm 2, \pm 1, \pm 1.5, \pm 0.5$
Plug these in to see which one gives you 0. Once you find one that does, then use synthetic division to divide your polynomial by ($x-$that factor). In this case, the factor that comes out is $x=-3$, so you would divide by (x+3).
Repeat until you've reduced to a quadratic, and then use the quadratic formula to find the irrational coefficients (if any).
Have you heard of the rational root test?
It implies that rational roots to this are of the form $p/q$ where $p$ divides $12$ and $q$ divides $2$. There are only a few numbers like this so you can check them.
Once you have a few roots, you can divide and get a smaller polynomial that is easier to manage.