Finding an explicit formula for a recursive sequence. How to show that the recurrent formula 
$$A_n=A_{n-1} + A_{n-2} +4.$$
gives a sequence of the form      $f(n)=cr^n+cr^n$?
The only way we are allowed to solve it, is with the quadratic formula $(r^2-r-1)$...

Consider $B_n=A_n+4$, to get $B_n=B_{n-1}+B_{n-2}$.
Now you can solve the chactaristic equation $r^2-r-1$.
This gives $r=\frac{1+\sqrt{5}}{2}$, $r=\frac{1-\sqrt{5}}{2}$.
Therefore we have $$B_n= A \left(\frac{1+\sqrt{5}}{2}\right)^n+ B \left(\frac{1-\sqrt{5}}{2}\right)^n$$
Therefore 
$$A_n= A \left(\frac{1+\sqrt{5}}{2}\right)^n+ B \left(\frac{1-\sqrt{5}}{2}\right)^n -4$$
Where $A,B$ depend on your initial conditions. 
