How to solve $x^3 + 2x + 2 \equiv 0 \pmod{25}$? My attempt was:
$x^3 + 2x + 2 \equiv 0 \pmod{25}$
By inspection, we see that $x \equiv 1 \pmod{5}$. is a solution of $x^3 + 2x + 2 \equiv 0 \pmod{5}$. Let $x = 1 + 5k$, then we have:
$$(1 + 5k)^3 + 2(1 + 5k) + 2 \equiv 0 \pmod{25}$$
$$\Leftrightarrow 125k^3 + 75k^2 + 25k + 5 \equiv 0 \pmod{25}$$ 
$$\Leftrightarrow 5 \equiv 0 \pmod{25}$$
And I was stuck here :( because k was completely cancelled out, how can we find solution for this equation?  
Thanks,

This shows, that there is no modular zero for $x^3+2x+2\equiv 0  \; (\text{mod}\; 25)$ of the form $x = 1 + 5k$, since clearly $ 5 \not\equiv 0 \; (\text{mod}\; 25)$.
Now go ahead and check $x = 3 + 5k$, because $x \equiv 3 \; (\text{mod}\; 5)$ is the other solution to $x^3+2x+2\equiv 0 \; (\text{mod}\; 5)$.
