Find the difference quotient of $LaTeX: \displaystyle f(x)=5 x^{3} + 8 x^{2} - 9 x - 8$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 9 h - 9 x + 5 \left(h + x\right)^{3} + 8 \left(h + x\right)^{2} - 8$ and expanding gives $LaTeX: \displaystyle f(x+h)=5 h^{3} + 15 h^{2} x + 8 h^{2} + 15 h x^{2} + 16 h x - 9 h + 5 x^{3} + 8 x^{2} - 9 x - 8$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(5 h^{3} + 15 h^{2} x + 8 h^{2} + 15 h x^{2} + 16 h x - 9 h + 5 x^{3} + 8 x^{2} - 9 x - 8)-(5 x^{3} + 8 x^{2} - 9 x - 8)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{5 h^{3} + 15 h^{2} x + 8 h^{2} + 15 h x^{2} + 16 h x - 9 h}{h}=5 h^{2} + 15 h x + 8 h + 15 x^{2} + 16 x - 9$