given the function f(x)=-x² - x - 12, find the difference quotient (f(x + h)-f(x))/h. (f(x + h)-f(x))/h =

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: [ \begin{align*} f(x + h)&=-(x + h)^2-(x + h)-12\ &=-(x^{2}+2xh+h^{2})-x - h-12\ &=-x^{2}-2xh - h^{2}-x - h-12 \end{align*} ]

Step2: Calculate $f(x + h)-f(x)$

[ \begin{align*} f(x + h)-f(x)&=(-x^{2}-2xh - h^{2}-x - h-12)-(-x^{2}-x - 12)\ &=-x^{2}-2xh - h^{2}-x - h-12 + x^{2}+x + 12\ &=-2xh - h^{2}-h \end{align*} ]

Step3: Find the difference - quotient

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{-2xh - h^{2}-h}{h}\ &=\frac{h(-2x - h - 1)}{h}\ &=-2x - h - 1 \end{align*} ]

Answer:

$-2x - h - 1$