given the following function, determine the difference quotient, $\frac{f(x + h) - f(x)}{h}$…

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: [ \begin{align*} f(x + h)&=6(x + h)^2+4(x + h)-3\ &=6(x^{2}+2xh+h^{2})+4x + 4h-3\ &=6x^{2}+12xh+6h^{2}+4x + 4h-3 \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(6x^{2}+12xh+6h^{2}+4x + 4h-3)-(6x^{2}+4x - 3)\ &=6x^{2}+12xh+6h^{2}+4x + 4h-3 - 6x^{2}-4x + 3\ &=12xh+6h^{2}+4h \end{align*} ]

Step3: Calculate the difference quotient

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{12xh+6h^{2}+4h}{h}\ &=\frac{h(12x + 6h+4)}{h}\ &=12x+6h + 4 \end{align*} ]

Answer:

$12x+6h + 4$