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

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=2 h + 2 x + 2 \left(h + x\right)^{3} - 9 \left(h + x\right)^{2} - 3$ and expanding gives $LaTeX: \displaystyle f(x+h)=2 h^{3} + 6 h^{2} x - 9 h^{2} + 6 h x^{2} - 18 h x + 2 h + 2 x^{3} - 9 x^{2} + 2 x - 3$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(2 h^{3} + 6 h^{2} x - 9 h^{2} + 6 h x^{2} - 18 h x + 2 h + 2 x^{3} - 9 x^{2} + 2 x - 3)-(2 x^{3} - 9 x^{2} + 2 x - 3)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{2 h^{3} + 6 h^{2} x - 9 h^{2} + 6 h x^{2} - 18 h x + 2 h}{h}=2 h^{2} + 6 h x - 9 h + 6 x^{2} - 18 x + 2$