Find the difference quotient of $LaTeX: \displaystyle f(x)=- 3 x^{2} + 10 x + 7$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=10 h + 10 x - 3 \left(h + x\right)^{2} + 7$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 3 h^{2} - 6 h x + 10 h - 3 x^{2} + 10 x + 7$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 3 h^{2} - 6 h x + 10 h - 3 x^{2} + 10 x + 7)-(- 3 x^{2} + 10 x + 7)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 3 h^{2} - 6 h x + 10 h}{h}=- 3 h - 6 x + 10$