Find the difference quotient of $LaTeX: \displaystyle f(x)=10 x^{3} - 8 x^{2} - 8 x - 4$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 8 h - 8 x + 10 \left(h + x\right)^{3} - 8 \left(h + x\right)^{2} - 4$ and expanding gives $LaTeX: \displaystyle f(x+h)=10 h^{3} + 30 h^{2} x - 8 h^{2} + 30 h x^{2} - 16 h x - 8 h + 10 x^{3} - 8 x^{2} - 8 x - 4$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(10 h^{3} + 30 h^{2} x - 8 h^{2} + 30 h x^{2} - 16 h x - 8 h + 10 x^{3} - 8 x^{2} - 8 x - 4)-(10 x^{3} - 8 x^{2} - 8 x - 4)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{10 h^{3} + 30 h^{2} x - 8 h^{2} + 30 h x^{2} - 16 h x - 8 h}{h}=10 h^{2} + 30 h x - 8 h + 30 x^{2} - 16 x - 8$