Answer
$P(a)=12$
Work Step by Step
Using the synthetic division shown below, the remainder (bottom-right number) of $(
x^3+7x^2+4x
)\div(
x+2
),$ is
\begin{align*}
12
.\end{align*}
Substituting $x=
-2
$ in $P(x)=
x^3+7x^2+4x
,$ then by the Remainder Theorem, the remainder when $P(x)$ is divided by $
x+2
$ is
\begin{align*}
P(-2)&=
(-2)^3+7(-2)^2+4(-2)
\\&=
-8+7(4)+4(-2)
\\&=
-8+28-8
\\&=
12
.\end{align*}
The two solutions above show that the remainder, $P(a),$ is
\begin{align*}
P(a)=12
.\end{align*}
