Find the difference quotient of $LaTeX: \displaystyle f(x)=- 8 x^{3} + 7 x^{2} - 6 x - 7$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 6 h - 6 x - 8 \left(h + x\right)^{3} + 7 \left(h + x\right)^{2} - 7$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 8 h^{3} - 24 h^{2} x + 7 h^{2} - 24 h x^{2} + 14 h x - 6 h - 8 x^{3} + 7 x^{2} - 6 x - 7$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 8 h^{3} - 24 h^{2} x + 7 h^{2} - 24 h x^{2} + 14 h x - 6 h - 8 x^{3} + 7 x^{2} - 6 x - 7)-(- 8 x^{3} + 7 x^{2} - 6 x - 7)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 8 h^{3} - 24 h^{2} x + 7 h^{2} - 24 h x^{2} + 14 h x - 6 h}{h}=- 8 h^{2} - 24 h x + 7 h - 24 x^{2} + 14 x - 6$