The number of distinct real roots of $x^4-4x^3+12x^2+x-1=0$

Let $f\left(x\right)=x^4-4x^3+12x^2+x-1$.

Clearly, it is an even degree whose last term is negative and coefficient of highest degree term is positive. So, it has at least two real roots one positive and one negative.

We have $f\left(0\right)=-1,f\left(1\right)=9$ and $f(-1)=15$

Thus, one real root lies between O and 1, and other lies between -1 and 0.

Now, $f^{\mathrm{\prime }}\left(x\right)=4x^3-12x^2+24x+1$ and $f^{\prime \prime }\left(x\right)=12\left(x^2-2x+2\right)$

Clearly, $f^{\prime \prime }\left(x\right)$ has no real root and $f^{\mathrm{\prime }}\left(x\right)$ being an odd degree polynomial, has at least one real root So, by the algebraic interpretation of Rolle's theorem $f\text{'} \left(x\right)$ has exactly one real root. Consequently,$f \left(x\right)$ has exactly two real roots.