The number of distinct real roots of $x^{4} - 4x^{3} + 12x^{2} + x - 1 = 0$ is ______.

The number of distinct real roots of x⁴ - 4x³ + 12x² + x - 1 = 0 is ______.

$f(x) = x^{4} - 4x^{3} + 12x^{2} + x - 1$

$\text{To find: the number of distinct real roots}$

$\text{sub-type: monotonicity / curve sketching}$

$1.\ \text{find } f'(x) \text{ and } f''(x)$

$2.\ \text{classify the one critical point, then count roots}$

$f(x) = x^{4} - 4x^{3} + 12x^{2} + x - 1$

$f'(x) = 4x^{3} - 12x^{2} + 24x + 1$

$f''(x) = 12x^{2} - 24x + 24$

$= 12\left[(x-1)^{2} + 1\right] > 0 \quad \forall x$

$f' \text{ strictly increasing} \Rightarrow f' \text{ has exactly one real zero } c$

$f'(0)=1>0,\ f'(-1)=-39<0 \Rightarrow c \in (-1,0)$

$f' < 0 \text{ for } x < c, \qquad f' > 0 \text{ for } x > c$

$f': -\to+ \text{ at } c \Rightarrow \text{local minimum}$

$f(-1) = 1+4+12-1-1 = 15 > 0$

$f(0) = -1 < 0, \quad f(1) = 1-4+12+1-1 = 9 > 0$

$f(-1)>0>f(0) \Rightarrow \text{one root in } (-1,0)$

$f(0)<0<f(1) \Rightarrow \text{one root in } (0,1)$

$\text{one turning point} \Rightarrow \text{exactly 2 real roots}$

$\boxed{2}$