Find the difference quotient of $LaTeX: \displaystyle f(x)=- 6 x^{2} + 5 x + 1$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=5 h + 5 x - 6 \left(h + x\right)^{2} + 1$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 6 h^{2} - 12 h x + 5 h - 6 x^{2} + 5 x + 1$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 6 h^{2} - 12 h x + 5 h - 6 x^{2} + 5 x + 1)-(- 6 x^{2} + 5 x + 1)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 6 h^{2} - 12 h x + 5 h}{h}=- 6 h - 12 x + 5$