Find the difference quotient of $LaTeX: \displaystyle f(x)=- 6 x^{2} + 3 x - 7$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=3 h + 3 x - 6 \left(h + x\right)^{2} - 7$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 6 h^{2} - 12 h x + 3 h - 6 x^{2} + 3 x - 7$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 6 h^{2} - 12 h x + 3 h - 6 x^{2} + 3 x - 7)-(- 6 x^{2} + 3 x - 7)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 6 h^{2} - 12 h x + 3 h}{h}=- 6 h - 12 x + 3$