$\text{ The curve }y=x^3+x^2-x\text{ has two horizontal tangents. The distance between these two horizontal lines is}$

$\begin{array}{l}\text{ Given that }f\left(x\right)=x^3+x^2-x=y\\ if the \tan gents are parallel to x-axis then \frac{dy}{dx}=0 \\ \Rightarrow 3x^2+2x-1=0 \\ \text{ ⇒ }x=-1,x=\frac{1}{3}\\ \text{ The tangents, at }x=-1,x=\frac{1}{3}\text{are }y=1,y=\frac{-5}{27}\end{array}$

$\therefore \text{ Distance between horizontal lines }=\left|y_1-y_2\right|=1+\frac{5}{27}=\frac{32}{27}$