How to show that $f(x)= x^4 + 2x^2 + x −1$ is irreducible over $\mathbb{R}$?
If its degree where odd, then obviously it must be reducible over $\mathbb{R}$, but here degree is even. I also do know its reducible over field with three elements. It's irreducible over $\mathbb{Z}_2$. So by $p$-modulo test it's irreducible over $\mathbb{Q}$. But I am getting annoyed when it comes to $\mathbb{R}$. Is there any short method to cope such problem in bond of times like within $1$ or $2$ minutes span of time.
I think this problem is solved if we just consider the comment of the user AnotherJohnDoe as above. But there is a another simple view to deal with it. Now let $f(x):=x^{4}+2x^{2}+x-1$ and hence this is a real-valued continuous function on the interval $[0,1]$ with $f(0)=-1$ and $f(1)=3.$ Thus, we see on account of $0$ is between $-1$ and $3$, one has that $f(\eta)=0$ for some $\eta\in(0,1),$ where we used the Intermediate value theorem to conclude. So $f$ has a factor $(x-\eta)$ with $\eta\in(0,1).$