Find the difference quotient of $LaTeX: \displaystyle f(x)=- x^{3} + 4 x^{2} + 2 x - 1$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=2 h + 2 x - \left(h + x\right)^{3} + 4 \left(h + x\right)^{2} - 1$ and expanding gives $LaTeX: \displaystyle f(x+h)=- h^{3} - 3 h^{2} x + 4 h^{2} - 3 h x^{2} + 8 h x + 2 h - x^{3} + 4 x^{2} + 2 x - 1$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- h^{3} - 3 h^{2} x + 4 h^{2} - 3 h x^{2} + 8 h x + 2 h - x^{3} + 4 x^{2} + 2 x - 1)-(- x^{3} + 4 x^{2} + 2 x - 1)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- h^{3} - 3 h^{2} x + 4 h^{2} - 3 h x^{2} + 8 h x + 2 h}{h}=- h^{2} - 3 h x + 4 h - 3 x^{2} + 8 x + 2$