Answer
$\dfrac{f(x+h)-f(x)}{h}=3x^{2}+3xh+h^{2}$
$\dfrac{f(x)-f(a)}{x-a}=x^{2}+ax+a^{2}$
Work Step by Step
$f(x)=x^{3}+2 $
$\textbf{Evaluate}$ $\dfrac{f(x+h)-f(x)}{h}$
First, substitute $x$ by $x+h$ in the given function and simplify to find $f(x+h)$:
$f(x+h)=(x+h)^{3}+2=x^{3}+3x^{2}h+3xh^{2}+h^{3}+2$
Substitute $f(x+h)$ and $f(x)$ into the difference quotient formula and simplify:
$\dfrac{f(x+h)-f(x)}{h}=\dfrac{x^{3}+3x^{2}h+3xh^{2}+h^{3}+2-(x^{3}+2)}{h}=...$
$...=\dfrac{x^{3}+3x^{2}h+3xh^{2}+h^{3}+2-x^{3}-2}{h}=...$
$...=\dfrac{3x^{2}h+3xh^{2}+h^{3}}{h}=...$
Take out common factor $h$ from the numerator and simplify again:
$...=\dfrac{h(3x^{2}+3xh+h^{2})}{h}=3x^{2}+3xh+h^{2}$
$\textbf{Evaluate}$ $\dfrac{f(x)-f(a)}{x-a}$
Substitute $x$ by $a$ in $f(x)$ to find $f(a)$:
$f(a)=a^{3}+2$
Substitute $f(a)$ and $f(x)$ into the difference quotient formula and simplify:
$\dfrac{f(x)-f(a)}{x-a}=\dfrac{x^{3}+2-(a^{3}+2)}{x-a}=\dfrac{x^{3}+2-a^{3}-2}{x-a}=...$
$...=\dfrac{x^{3}-a^{3}}{x-a}=...$
Factor the numerator and simplify again:
$...=\dfrac{(x-a)(x^{2}+ax+a^{2})}{x-a}=x^{2}+ax+a^{2}$
