I want to find solutions to the equation $x^3 + 2x - 3 \equiv 0 (mod 45)$. I have already found solutions to $x^3 + 2x - 3 \equiv 0 (mod 5)$ and $x^3 + 2x - 3 \equiv 0 (mod 9)$, simply by brute force. Respectively, $x = 1, 3 (mod 5)$ and $x = 1, 2, 6 (mod 9)$ are zeros of the equations. However, the brute force method doesn't seem like a wise way to approach this problem for $mod 45$.
I'm aware of the CRT, I'm just unsure how to apply that here if applicable. This is for homework, so I'd prefer a hint or something to point me in the right direction rather than an answer.