Answer
$\text{f o g}(x)=x^2+4x+5$
$\text{g o f}(x)=x^2+3$
Work Step by Step
Given $g: \cal{R}\rightarrow \cal{R}$ and $f: \cal{R}\rightarrow \cal{R}$
$$f(x)=x^2+1$$$$g(x)=x+2$$
Since f and g are both functions from $\cal{R}$ to $\cal{R}$, $\text{f o g}$ and $\text{g o f}$ are also functions from $\cal{R}$ to $\cal{R}$.
Use the definition of composition:
$$\text{f o g}(x)=f(g(x))=fxX+2)=(x+2)^2+1=x^2+4x+5$$$$\text{g o f}(x)=g(f(x))=g(x^2+1)=(x^2+1)+1=x^2+3$$
