Find the difference quotient of $LaTeX: \displaystyle f(x)=7 x^{3} - 2 x^{2} - 2 x + 4$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 2 h - 2 x + 7 \left(h + x\right)^{3} - 2 \left(h + x\right)^{2} + 4$ and expanding gives $LaTeX: \displaystyle f(x+h)=7 h^{3} + 21 h^{2} x - 2 h^{2} + 21 h x^{2} - 4 h x - 2 h + 7 x^{3} - 2 x^{2} - 2 x + 4$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(7 h^{3} + 21 h^{2} x - 2 h^{2} + 21 h x^{2} - 4 h x - 2 h + 7 x^{3} - 2 x^{2} - 2 x + 4)-(7 x^{3} - 2 x^{2} - 2 x + 4)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{7 h^{3} + 21 h^{2} x - 2 h^{2} + 21 h x^{2} - 4 h x - 2 h}{h}=7 h^{2} + 21 h x - 2 h + 21 x^{2} - 4 x - 2$