College Algebra, Chapter 3, Review Exercises, Section Review Exercises, Problem 58

Find the average rate of change of the function $f(x) = (x+1)^2$ between $x = a$ and $x = a+h$
Recall that the formula for the average rate is.
$\displaystyle \frac{f(b) - f(a)}{b - a}$

$\begin{equation} \begin{aligned} \frac{f(b) - f(a)}{b - a} &= \frac{f(a+h)-f(a)}{(a+h)-a} = \frac{(a+h+1)^2-(a+1)^2}{h}\\ \\ &= \frac{\left[a^2 + ah + a + ah + h^2 + h + a + h + 1 \right]- \left[ a^2 + 2a + 1 \right]}{h}\\ \\ &= \frac{a^2 + 2ah + 2a + h^2+ 2h + 1 - a^2 - 2a - 1}{h}\\ \\ &= \frac{2ah + h^2 + 2h}{h}\\ \\ &= \frac{h(2a + h + 2)}{h}\\ \\ &= 2a + h + 2 \\ \\ &= 2(a+1) + h \end{aligned} \end{equation}$