How do I solve this equation; $x^4+4x-1=0$? I know that I can easly solve this with the $4$th degree's equation, but isn't there a smarter way? It is an olympiad's problem so it shouldn't be a formula, but more find a formula... I have remarked that it is equal to $(x^2+1)\cdot(x+1)\cdot(x-1)+4x=0$ It isn't just solving the equation, I want to know if it can be solved otherwise than that... Could someone explain why this is equal to $(x^2+1)^2-(\sqrt{2}(x-1))^2$ please ?

If you don't want to rely on tricks (tricks are always useful though), a more systematic approach is as follows. Let
$$x^4+4x-1 = (x^2+ax+b)(x^2+cx+d)$$
We then have
\begin{align*}
a+c & = 0\\
b+d+ac & = 0\\
bc + ad & = 4\\
bd & = -1
\end{align*}
Eliminating $c$, we obtain
\begin{align*}
b+d & = a^2\\
d-b & = \dfrac4{a}\\
bd & = -1
\end{align*}
This gives us $d = \dfrac{a^2}2 + \dfrac2a$ and $b = \dfrac{a^2}2 - \dfrac2a$. Since $bd=-1$, we obtain
$$\dfrac{a^4}4 - \dfrac4{a^2} = -1 \implies a^6 - 16 = -4a^2$$
Setting $a^2=y$, we have
$$y^3+4y - 16 = 0$$
By rational root theorem, we find that $y=2$ as one root. The other two roots are complex and hence we are not interested in them. This gives us $a=\pm \sqrt2$. Taking $a=\sqrt2$, we see $c=-\sqrt2$. This gives us
$$d=1+\sqrt2 \text{ and }b = 1-\sqrt2$$
Hence, we obtain the factorization as
$$x^4+4x-1 = (x^2+\sqrt2 x+1-\sqrt2)(x^2-\sqrt2 x+1+\sqrt2)$$
