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,

Hint by others, we use Hensel's lemma. By Hensel's lemma we solve $x^{3}+2x+2=0$ (mod 5). This has solution $1$ and $3$. But $1$ satisfies $f'(1)=0$, by Chan's previous work it should be cast aside. Hensel's lemma suggests solutions of the form $3+5k$ with $3(3+5k)^{2}+2\not=0$(mod 5). Hensel's lemma give $k$ to be:
$k(29+90 k+75 k^2)=k^{3}+2k+2$ (mod 5)
Solve this we should get $k=2$. I do not know if this is the best way to solve it. 
