Answer
$(fg)(x)=-38$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To evaluate the given expression, $
(fg)(4)
,$ given
\begin{array}{l}\require{cancel}
f(x)=x^2+3
\\
g(x)=-2x+6
,\end{array}
use the definition of the appropriate function operation. Then substitute $x$ with $
4
.$
$\bf{\text{Solution Details:}}$
Since $(fg)(x)=f(x)g(x),$ then
\begin{array}{l}\require{cancel}
(fg)(x)=(x^2+3)(-2x+6)
\\\\
(fg)(x)=x^2(-2x)+x^2(6)+3(-2x)+3(6)
\\\\
(fg)(x)=-2x^3+6x^2-6x+18
.\end{array}
Substituting $x$ with $
4
,$ then
\begin{array}{l}\require{cancel}
(fg)(x)=-2(4)^3+6(4)^2-6(4)+18
\\\\
(fg)(x)=-2(64)+6(16)-6(4)+18
\\\\
(fg)(x)=-128+96-24+18
\\\\
(fg)(x)=-38
.\end{array}