I've been thinking for a while about how to go about solving the equation $x^2+3x+8 \equiv 0 \pmod{144}$ and similar ones. When the modulus is odd it's not too tricky, but when it is I can't see quite how to find all solutions. I tried solving $x^2+3x+8 \equiv 0 \pmod{9}$ ($x=2$ or 4) and $x^2+3x+8\equiv 0 \pmod{16}$ ($x=5$ or 8) and then using the CRT to find the solutions, $x=40, 56, 85, 101$, which also solve the original equation. Brute forcing shows that there are no more solutions, but this method seems suboptimal. Is there a systematic way to solve this type of equation?
Edit: Apparently I made a mistake. Solving the equation in mod 16 and 27 on their own and then using the CRT gives all 4 solutions. However, I would still be interested to know if there is a better way! (for instance, now I solved $x^2+3x+8 \equiv 0 \pmod{16}$ by force, which isn't too difficult but seems unneccesary).