List questions from polynomial-congruences

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Need hints on the following algebra problems.

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Let $R = \mathbb{Q}[x]$ and let $I = (x^2 + 2x + 2)R$ be the principal ideal generated by $x^2 + 2x +2$. Two questions are below.

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List Question

solve $x^2 -4x +13 \equiv 0 \pmod{81}$?

Equality of polynomial functions modulo n

Need hints on the following algebra problems.

What is the congruence class of $x^3\mod x^3+x+1$?

How to find all the roots in this ring?

Solution of polynomial congruence

Let $R = \mathbb{Q}[x]$ and let $I = (x^2 + 2x + 2)R$ be the principal ideal generated by $x^2 + 2x +2$. Two questions are below.

Solutions of $x^2-6x-13 \equiv 0 \pmod{127}$

Can I used polynomial congruence for prove 3 divide $n^3-n$?

How to find the multiplicity of a polynomial?

Find $f\in \mathbb{Z}_{p}\left[X\right]$ such that $f\left(X^{m}\right)$ is divisible by $\Phi_{p-1}$.

Most efficient solution to find polynomial congruence for 0 mod p

Show that a congruence involving prime power is solvable.

How many quadratic functions mod 12 have exactly two roots?

Find coefficient $c$ that makes a valid congruences

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