Find the difference quotient of $LaTeX: \displaystyle f(x)=5 x^{3} + 2 x^{2} - 6 x + 10$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 6 h - 6 x + 5 \left(h + x\right)^{3} + 2 \left(h + x\right)^{2} + 10$ and expanding gives $LaTeX: \displaystyle f(x+h)=5 h^{3} + 15 h^{2} x + 2 h^{2} + 15 h x^{2} + 4 h x - 6 h + 5 x^{3} + 2 x^{2} - 6 x + 10$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(5 h^{3} + 15 h^{2} x + 2 h^{2} + 15 h x^{2} + 4 h x - 6 h + 5 x^{3} + 2 x^{2} - 6 x + 10)-(5 x^{3} + 2 x^{2} - 6 x + 10)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{5 h^{3} + 15 h^{2} x + 2 h^{2} + 15 h x^{2} + 4 h x - 6 h}{h}=5 h^{2} + 15 h x + 2 h + 15 x^{2} + 4 x - 6$