I would like to show that polynomial $x^3-xy^2+y+1$ is irreducible over $\mathbb{Q}(x)[y]$. I thought I could find the roots in using $\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$, but I found that the two roots are $\frac{-1±\sqrt{4x^4+4x+1}}{-2x}$. These two roots couldn't be in $\mathbb{Q}(x)$.
Is anyone could help me to solve this problem?
You are correct: $4 x^4 + 4 x + 1$ does not have a square root in $\mathbb Q(x)$, and so your polynomial is irreducible. If $\sqrt{4 x^4 + 4 x + 1}$ was a rational function of $x$, it would have to be a quadratic polynomial. But the Maclaurin series of $\sqrt{4 x^4 + 4 x + 1}$ starts $1 + 2 x - 2 x^2 + 4 x^3 + \ldots$, so it isn't.