find the difference quotient (f(x + h)-f(x))/h, where h≠0, for the function below. f(x)=-4x² - x + 6…

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: [ \begin{align*} f(x + h)&=-4(x + h)^2-(x + h)+6\ &=-4(x^{2}+2xh+h^{2})-x - h+6\ &=-4x^{2}-8xh-4h^{2}-x - h+6 \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(-4x^{2}-8xh-4h^{2}-x - h+6)-(-4x^{2}-x + 6)\ &=-4x^{2}-8xh-4h^{2}-x - h+6 + 4x^{2}+x - 6\ &=-8xh-4h^{2}-h \end{align*} ]

Step3: Find the difference - quotient

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{-8xh-4h^{2}-h}{h}\ &=\frac{h(-8x-4h - 1)}{h}\ &=-8x-4h - 1 \end{align*} ]

Answer:

$-8x-4h - 1$