[Community Question] Algebra-Precalculus: Find the remainder of the division of $x^n+5$ with $x^3+10x^2+25x$

One of our user asked:

Find the remainder of the division of $x^n+5$ with $x^3+10x^2+25x$ over $\mathbb{Q}$

What I tried to do is to write $x^n+5=p(x)(x^3+10x^2+25x)+Ax^2+Bx+C$, where $p(x)$ is a polynomial of degree $n-3$. If I set $x=0$ I obtain that $C=0$.

Now $x^3+10x^2+25x=x(x+5)^2$ and by setting $x=-5$, I get that

$5A-B=(-1)^n5^{n-1}$

But I need one more equation to be able to find the coefficients of the remainder, and I can't get one. What should I do?

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