Answer
$\text{a) }
f(g(x))=4x^2-12x+41
\\\\\text{b) }
g(f(x))=16x^2-68x+82$
Work Step by Step
$\bf{\text{Solution Outline:}}$
With
\begin{array}{l}\require{cancel}
f(x)=
4x-7
\\g(x)=
x^2-3x+12
,\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(x^2-3x+12)
\\\\
f(g(x))=4(x^2-3x+12)-7
\\\\
f(g(x))=4x^2-12x+48-7
\\\\
f(g(x))=4x^2-12x+41
.\end{array}
Replacing $x$ with $f(x)$ in $g.$ Hence,
\begin{array}{l}\require{cancel}
g(f(x))=g(4x-7)
\\\\
g(f(x))=(4x-7)^2-3(4x-7)+12
\\\\
g(f(x))=(16x^2-56x+49)+(-12x+21)+12
\\\\
g(f(x))=16x^2-68x+82
.\end{array}
Hence,
\begin{array}{l}\require{cancel}
\text{a) }
f(g(x))=4x^2-12x+41
\\\\\text{b) }
g(f(x))=16x^2-68x+82
.\end{array}
