Find the difference quotient of $LaTeX: \displaystyle f(x)=- 5 x^{3} - 7 x^{2} + 4 x + 5$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=4 h + 4 x - 5 \left(h + x\right)^{3} - 7 \left(h + x\right)^{2} + 5$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 5 h^{3} - 15 h^{2} x - 7 h^{2} - 15 h x^{2} - 14 h x + 4 h - 5 x^{3} - 7 x^{2} + 4 x + 5$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 5 h^{3} - 15 h^{2} x - 7 h^{2} - 15 h x^{2} - 14 h x + 4 h - 5 x^{3} - 7 x^{2} + 4 x + 5)-(- 5 x^{3} - 7 x^{2} + 4 x + 5)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 5 h^{3} - 15 h^{2} x - 7 h^{2} - 15 h x^{2} - 14 h x + 4 h}{h}=- 5 h^{2} - 15 h x - 7 h - 15 x^{2} - 14 x + 4$