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Finding number of solutions for $f(x)=x^4+4x^3+17x^2+10x+18$ given $f(x)=0$

Given a polynomial $f(x)=x^4+4x^3+17x^2+10x+18$. I have to find the number of solutions for $f(x)=0$. I thought of putting $f(x)=x^4(x^2+4x+11)$. But what to do after that? I would be thankful if someone could give me a clue to get out of this.

A:

Since $f(x)$ is irreducible over $\mathbb Q$, there is only one solution, namely $x = 0$.

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