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

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 5 h - 5 x + 2 \left(h + x\right)^{3} + 6 \left(h + x\right)^{2} - 8$ and expanding gives $LaTeX: \displaystyle f(x+h)=2 h^{3} + 6 h^{2} x + 6 h^{2} + 6 h x^{2} + 12 h x - 5 h + 2 x^{3} + 6 x^{2} - 5 x - 8$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(2 h^{3} + 6 h^{2} x + 6 h^{2} + 6 h x^{2} + 12 h x - 5 h + 2 x^{3} + 6 x^{2} - 5 x - 8)-(2 x^{3} + 6 x^{2} - 5 x - 8)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{2 h^{3} + 6 h^{2} x + 6 h^{2} + 6 h x^{2} + 12 h x - 5 h}{h}=2 h^{2} + 6 h x + 6 h + 6 x^{2} + 12 x - 5$