given the following function, determine the difference quotient, $\frac{f(x + h)-f(x)}{h}$. $f(x)=4x^{2}-8x…

Explanation:

Step1: Find $f(x + h)$

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

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

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

Step3: Find the difference - quotient $\frac{f(x + h)-f(x)}{h}$

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

Answer:

$8x + 4h-8$