Sophie Germain Identity?

Let $p(x) = x^4 - 4x + 1 \Rightarrow p'(x) = 4x^3 - 4$ which is an increasing strictly polynomial with $\displaystyle\lim_{x \to -\infty} p'(x) = - \infty$ and $\displaystyle\lim_{x \to \infty} p'(x) = \infty$ . This tell us that the number of real solutions of $p(x)$ is at most $2$ . Furthemore, because of $p(x)$ is a polynomial of $4^{th}$ degree with real coefficients, the number of real solutions is even(in this case it will be 0 or 2), together to $p(1) < 0$ and $p(2) >0$ and applying intermediate value theorem implies that the number of real solutions is $2$

apply descarte's rule

(NMTC inter 2016)