Find the difference quotient of $LaTeX: \displaystyle f(x)=10 x^{3} + 7 x^{2} - 9 x - 7$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 9 h - 9 x + 10 \left(h + x\right)^{3} + 7 \left(h + x\right)^{2} - 7$ and expanding gives $LaTeX: \displaystyle f(x+h)=10 h^{3} + 30 h^{2} x + 7 h^{2} + 30 h x^{2} + 14 h x - 9 h + 10 x^{3} + 7 x^{2} - 9 x - 7$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(10 h^{3} + 30 h^{2} x + 7 h^{2} + 30 h x^{2} + 14 h x - 9 h + 10 x^{3} + 7 x^{2} - 9 x - 7)-(10 x^{3} + 7 x^{2} - 9 x - 7)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{10 h^{3} + 30 h^{2} x + 7 h^{2} + 30 h x^{2} + 14 h x - 9 h}{h}=10 h^{2} + 30 h x + 7 h + 30 x^{2} + 14 x - 9$