Answer
$\text{a) }
f(g(x))=4x^2+x+3
\\\\\text{b) }
g(f(x))=4x^2+17x+19$
Work Step by Step
$\bf{\text{Solution Outline:}}$
With
\begin{array}{l}\require{cancel}
f(x)=
x+2
\\g(x)=
4x^2+x+1
,\end{array}
replace $x$ with $g(x)$ in $f$ to find $f(g(x)).$ To find $g(f(x)),$ replace $x$ with $f(x)$ in $g.$
$\bf{\text{Solution Details:}}$
Replacing $x$ with $g(x)$ in $f,$ then
\begin{array}{l}\require{cancel}
f(g(x))=f(4x^2+x+1)
\\\\
f(g(x))=(4x^2+x+1)+2
\\\\
f(g(x))=4x^2+x+3
.\end{array}
Replacing $x$ with $f(x)$ in $g.$ Hence,
\begin{array}{l}\require{cancel}
g(f(x))=g(x+2)
\\\\
g(f(x))=4(x+2)^2+(x+2)+1
\\\\
g(f(x))=4(x^2+4x+4)+(x+2)+1
\\\\
g(f(x))=(4x^2+16x+16)+(x+2)+1
\\\\
g(f(x))=4x^2+17x+19
.\end{array}
Hence,
\begin{array}{l}\require{cancel}
\text{a) }
f(g(x))=4x^2+x+3
\\\\\text{b) }
g(f(x))=4x^2+17x+19
.\end{array}