<s>[INST] Finding zero divisors in a polynomial quotient ring 
Is $x^2+x+I$ a zero divisor in $\mathbb{Z}_7[x]/I$, where $I=(x^3+5x^2+2x+5)$?

I know that $\gcd(x^3+5x^2+2x+5,x^2+x)=x+1$, and that it means that $x^2+x$ is indeed a zero divisor. 
What I struggle with is finding $g(x)$ such that $$(f(x)+\langle x^3+5x^2+2x+5\rangle)(g(x)+\langle x^3+5x^2+2x+5\rangle)=\langle x^3+5x^2+2x+5\rangle$$ in the quotient ring.
I know how to find an inverse, is this any similar? How is it done?
 [/INST] Hint:
Note that $6$ is a root of given polynomial. That is, in $\Bbb Z_7$,  $$x^3+5x^2+2x+5=(x-6)(x^2+4x+5)=(x+1)(x^2+4x+5)$$ So take $f(x)=x+1$ and $g(x)=x^2+4x+5$ and let $I=\langle x^3+5x^2+2x+5 \rangle$. Then $f+I \neq I$ and $g+I \neq I$ but $$(f+I)(g+I)=fg+I=I$$
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