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

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 3 h - 3 x + 6 \left(h + x\right)^{3} + 2 \left(h + x\right)^{2} + 7$ and expanding gives $LaTeX: \displaystyle f(x+h)=6 h^{3} + 18 h^{2} x + 2 h^{2} + 18 h x^{2} + 4 h x - 3 h + 6 x^{3} + 2 x^{2} - 3 x + 7$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(6 h^{3} + 18 h^{2} x + 2 h^{2} + 18 h x^{2} + 4 h x - 3 h + 6 x^{3} + 2 x^{2} - 3 x + 7)-(6 x^{3} + 2 x^{2} - 3 x + 7)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{6 h^{3} + 18 h^{2} x + 2 h^{2} + 18 h x^{2} + 4 h x - 3 h}{h}=6 h^{2} + 18 h x + 2 h + 18 x^{2} + 4 x - 3$