How to solve a recursive equation

How to solve a recursive equation

I have been given a task to solve the following recursive equation
\begin{align*} a_1&=-2\\ a_2&= 12\\ a_n&= -4a_n{}_-{}_1-4a_n{}_-{}_2,
\quad n \geq 3. \end{align*}
Should I start by rewriting $a_n$ or is there some kind of approach to
solve these?
I tried rewriting it to a Quadratic Equation (English isn't my native
language, sorry if this is incorrect). Is this the right approach, if so
how do I continue?
\begin{align*} a_n&= -4a_n{}_-{}_1-4a_n{}_-{}_2\\ x^2&= -4x-4\\ 0&= x^2 +
4x + 4 \end{align*}