3. find the difference quotient (f(x + h)-f(x))/h and simplify. f(x)=-x^2 - 5x - 6

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: [ \begin{align*} f(x + h)&=-(x + h)^2-5(x + h)-6\ &=-(x^{2}+2xh+h^{2})-5x-5h - 6\ &=-x^{2}-2xh - h^{2}-5x-5h-6 \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(-x^{2}-2xh - h^{2}-5x-5h-6)-(-x^{2}-5x - 6)\ &=-x^{2}-2xh - h^{2}-5x-5h-6 + x^{2}+5x + 6\ &=-2xh - h^{2}-5h \end{align*} ]

Step3: Calculate the difference - quotient

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{-2xh - h^{2}-5h}{h}\ &=\frac{h(-2x - h - 5)}{h}\ &=-2x - h - 5 \end{align*} ]

Answer:

$-2x - h - 5$