(12)  Try to explain the monotonic increasing interval of the function $f(x)=\frac{(x+1)^{2}}{x^{2}+1}$.
Solution: Let $x \neq 0, f(x)=1+\frac{2 x}{x^{2}+1}=1+\frac{2}{x+\frac{1}{x}}$, and when $x>0$, $g(x)=x+$ $\frac{1}{x}>0$ and is monotonically decreasing on $(0,1]$, and monotonically increasing on $[1,+\infty)$, thus $f_{1}(x)=\frac{2}{x+\frac{1}{x}}$ is monotonically increasing on $(0,1]$, and monotonically decreasing on $[1,+\infty)$. Also, $f_{1}(x)=\frac{2 x}{x^{2}+1}$ is an odd function that is meaningful at $x=0$, so $f_{1}(x)=\frac{2 x}{x^{2}+1}$ is monotonically increasing on $[-1,1]$. The graph of $f(x)=\frac{(x+1)^{2}}{x^{2}+1}$ can be obtained by shifting the graph of $f_{1}(x)=\frac{2 x}{x^{2}+1}$ up by one unit, so the monotonically increasing interval of $f(x)=\frac{(x+1)^{2}}{x^{2}+1}$ is $[-1,1]$.

Solution Guide: Transform a relatively complex function into a common function, and comprehensively apply various properties of the function (including function graph transformations), to obtain a conclusion about a certain property of the function. This indicates that the various properties of a function are an indivisible whole, they influence each other, and we must examine a certain property of the function from multiple different perspectives.