Find the difference quotient of $LaTeX: \displaystyle f(x)=5 x^{3} - 6 x^{2} - 5 x + 4$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 5 h - 5 x + 5 \left(h + x\right)^{3} - 6 \left(h + x\right)^{2} + 4$ and expanding gives $LaTeX: \displaystyle f(x+h)=5 h^{3} + 15 h^{2} x - 6 h^{2} + 15 h x^{2} - 12 h x - 5 h + 5 x^{3} - 6 x^{2} - 5 x + 4$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(5 h^{3} + 15 h^{2} x - 6 h^{2} + 15 h x^{2} - 12 h x - 5 h + 5 x^{3} - 6 x^{2} - 5 x + 4)-(5 x^{3} - 6 x^{2} - 5 x + 4)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{5 h^{3} + 15 h^{2} x - 6 h^{2} + 15 h x^{2} - 12 h x - 5 h}{h}=5 h^{2} + 15 h x - 6 h + 15 x^{2} - 12 x - 5$