Common tangent

$\begin{aligned} P_1 &: y = x^2 - ( 6 + 2 \sqrt{5} ) x + ( 22 + 6 \sqrt{5} ) \\ P_2 &: y = -x^2 + ( 6 + 2 \sqrt{5} ) x - ( 6 + 6 \sqrt{5} ) \end{aligned}$

If the equation of the common tangent of the two parabolas $P_1$ and $P_2$ above is given by $L : ax+by+c=0$ , where $a,b \ge 0$ and $|b|$ and $|c|$ are coprime, find $a+b+c$ .