Find the difference quotient of $LaTeX: \displaystyle f(x)=5 x^{3} - 8 x^{2} + 8 x - 6$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=8 h + 8 x + 5 \left(h + x\right)^{3} - 8 \left(h + x\right)^{2} - 6$ and expanding gives $LaTeX: \displaystyle f(x+h)=5 h^{3} + 15 h^{2} x - 8 h^{2} + 15 h x^{2} - 16 h x + 8 h + 5 x^{3} - 8 x^{2} + 8 x - 6$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(5 h^{3} + 15 h^{2} x - 8 h^{2} + 15 h x^{2} - 16 h x + 8 h + 5 x^{3} - 8 x^{2} + 8 x - 6)-(5 x^{3} - 8 x^{2} + 8 x - 6)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{5 h^{3} + 15 h^{2} x - 8 h^{2} + 15 h x^{2} - 16 h x + 8 h}{h}=5 h^{2} + 15 h x - 8 h + 15 x^{2} - 16 x + 8$