We have $P(x) = x^2 + 3x + 9 ≡ 0 \mod(3^i)$, for $i=1,2,3,4$

for $\mod(3): x^2 + 3x + 9 ≡ x^2$, so $x ≡ 0 \mod(3)$ is a solution.

From Hensel lifting, we have $s = x + a\times 7,\ a = [-P(0)\times (1/3)] \times (P'(0))^{-1}$.
Computing for $a$, we get $a = -3 \times (3)^{-1}$. But, $(3)^{-1}$ is when $3a ≡ 1 \mod(3)$, but this has no solutions since $\gcd(3,3)$ is not $1$. Where am I going wrong? Can somebody help me out please? :(