Find the difference quotient of $LaTeX: \displaystyle f(x)=8 x^{3} + 6 x^{2} - 7 x - 2$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 7 h - 7 x + 8 \left(h + x\right)^{3} + 6 \left(h + x\right)^{2} - 2$ and expanding gives $LaTeX: \displaystyle f(x+h)=8 h^{3} + 24 h^{2} x + 6 h^{2} + 24 h x^{2} + 12 h x - 7 h + 8 x^{3} + 6 x^{2} - 7 x - 2$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(8 h^{3} + 24 h^{2} x + 6 h^{2} + 24 h x^{2} + 12 h x - 7 h + 8 x^{3} + 6 x^{2} - 7 x - 2)-(8 x^{3} + 6 x^{2} - 7 x - 2)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{8 h^{3} + 24 h^{2} x + 6 h^{2} + 24 h x^{2} + 12 h x - 7 h}{h}=8 h^{2} + 24 h x + 6 h + 24 x^{2} + 12 x - 7$