Find the difference quotient of $LaTeX: \displaystyle f(x)=8 x^{3} + 3 x^{2} - 8 x - 9$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 8 h - 8 x + 8 \left(h + x\right)^{3} + 3 \left(h + x\right)^{2} - 9$ and expanding gives $LaTeX: \displaystyle f(x+h)=8 h^{3} + 24 h^{2} x + 3 h^{2} + 24 h x^{2} + 6 h x - 8 h + 8 x^{3} + 3 x^{2} - 8 x - 9$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(8 h^{3} + 24 h^{2} x + 3 h^{2} + 24 h x^{2} + 6 h x - 8 h + 8 x^{3} + 3 x^{2} - 8 x - 9)-(8 x^{3} + 3 x^{2} - 8 x - 9)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{8 h^{3} + 24 h^{2} x + 3 h^{2} + 24 h x^{2} + 6 h x - 8 h}{h}=8 h^{2} + 24 h x + 3 h + 24 x^{2} + 6 x - 8$