Christmas Streak 32/88: Polynomial But Strange

Algebra Level 3

$P(x)$ is a cubic polynomial, and for $x=1,~2,~3,~4,$ $P(x)=\frac{1}{1+x+x^2}.$

For some positive coprime integers $a$ and $b,$ $P(5)=-\frac{a}{b}.$

Find the value of $a+b.$

This problem is a part of <Christmas Streak 2017> series.