Solve recurrence relation $a(n)=2\cdot a(n-1)+5\cdot a(n-2)+6\cdot a(n-3)$ and the associated cubic

I am trying to solve following :

$$a(n)=2\cdot a(n-1)+5\cdot a(n-2)+6\cdot a(n-3)$$ with the initial conditions given by $a(0)=3,a(1)=2,a(2)=14$.

So first of all, I want to mark that there exists a theorem which states the following:

Let $P(x)=u_n\cdot x^n+u_{n-1}\cdot x^{n-1}+....+u_1\cdot x+u_0$ be a polynomial with real coefficients, and $u_n\neq 0$. If $P(x)$ has rational roots, they are of the form $\pm p/q$ where $p|u_0$ and $q|u_n$.

So first what I have tried is to construct characteristic equation or $$r^3-2\cdot r^2-5\cdot r-6=0$$ Now I have a problem to solve cubic equation also from the theorem I need a few help to understand it well.