College Algebra, Chapter 3, 3.1, Section 3.1, Problem 36

Given the function $f(x) = x^2 +1$. Find $f(a)$, $f(a+h)$ and the difference quotient $\displaystyle \frac{f(a+h) - f(a)}{h}$ where $h \neq 0$
For $f(a)$
$f(a) = a^2 +1 $ Replace $x$ by $a$

For $f(a+h)$

$
\begin{equation}
\begin{aligned}
f(a+h) &= (a+h)^2 + 1 && \text{Replace } x \text{ by } (a+h)\\
\\
&= a^2 + 2ah + h^2 +1
\end{aligned}
\end{equation}
$


For $\displaystyle \frac{f(a+h)-f(a)}{h}$

$
\begin{equation}
\begin{aligned}
\frac{f(a-h)-f(a)}{h} &= \frac{(a+h)^2 + 1 - (a^2 + 1)}{h}\\
\\
&= \frac{a^2 + 2ah + h^2 + 1 - a^2 - 1}{h} && \text{Combine like terms}\\
\\
&= \frac{2ah + h^2}{h} && \text{Factor out } h \text{ from each term}\\
\\
&= \frac{\cancel{h}(2a + h)}{\cancel{h}} && \text{Cancel common factor}\\
\\
&= 2a + h
\end{aligned}
\end{equation}
$