discrete mathematics - System of congruences with polynomials

How do I go about solving exercises such as this one:

Find all polynomials $f(x)$ in $\mathbb{Z}_3$ that satisfy

$$f(x) \equiv 1 \space \space \mathrm{mod} \space \space x^2 + 1$$
$$f(x) \equiv x \space \space \mathrm{mod} \space \space x^3 + 2x + 2$$

in $\mathbb{Z}_3.$

I know about the Chinese Remainder Theorem, but only how to apply it to system of congruences where there are no polynomials involved.

I realise that $f_1(x) \equiv f_2(x) \space \space \mathrm{mod} \space g(x)$ means that $f_1(x) - f_2(x)$ is divisible by $g(x)$, but that's about as far as I've come with this problem.

Also, if anyone has any advice as to where I can read about modular arithmetic involving polynomials, I'd be happy to hear about it, because the literature I have doesn't say much about it at all, and I would like to learn.