find the difference quotient and simplify.\n$f(x)=-3x^{2}-8x + 2$\nthe difference quotient of $f(x)$ is \n

Explanation:

Step1: Recall difference quotient formula

The difference quotient of a function $f(x)$ is $\frac{f(x + h)-f(x)}{h}$, where $h\neq0$. First, find $f(x + h)$. Given $f(x)=- 3x^{2}-8x + 2$, then $f(x + h)=-3(x + h)^{2}-8(x + h)+2$. Expand $(x + h)^{2}=x^{2}+2xh+h^{2}$. So $f(x + h)=-3(x^{2}+2xh + h^{2})-8(x + h)+2=-3x^{2}-6xh-3h^{2}-8x-8h + 2$.

Step2: Substitute into difference - quotient formula

$\frac{f(x + h)-f(x)}{h}=\frac{(-3x^{2}-6xh-3h^{2}-8x-8h + 2)-(-3x^{2}-8x + 2)}{h}$. Simplify the numerator: [ \begin{align*} &(-3x^{2}-6xh-3h^{2}-8x-8h + 2)-(-3x^{2}-8x + 2)\ =&-3x^{2}-6xh-3h^{2}-8x-8h + 2 + 3x^{2}+8x-2\ =&-6xh-3h^{2}-8h \end{align*} ]

Step3: Simplify the quotient

$\frac{-6xh-3h^{2}-8h}{h}=\frac{h(-6x - 3h-8)}{h}=-6x-3h - 8$.

Answer:

$-6x-3h - 8$