Find the difference quotient of $LaTeX: \displaystyle f(x)=- 10 x^{3} - 8 x^{2} + 7 x + 1$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=7 h + 7 x - 10 \left(h + x\right)^{3} - 8 \left(h + x\right)^{2} + 1$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 10 h^{3} - 30 h^{2} x - 8 h^{2} - 30 h x^{2} - 16 h x + 7 h - 10 x^{3} - 8 x^{2} + 7 x + 1$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 10 h^{3} - 30 h^{2} x - 8 h^{2} - 30 h x^{2} - 16 h x + 7 h - 10 x^{3} - 8 x^{2} + 7 x + 1)-(- 10 x^{3} - 8 x^{2} + 7 x + 1)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 10 h^{3} - 30 h^{2} x - 8 h^{2} - 30 h x^{2} - 16 h x + 7 h}{h}=- 10 h^{2} - 30 h x - 8 h - 30 x^{2} - 16 x + 7$