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

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