Answer
$P(a)=51$
Work Step by Step
Using the synthetic division below, the remainder (bottom-right number) of $(
2x^3+4x^2-10x-9
)\div\left(
x-3
\right)$ is
\begin{align*}
51
.\end{align*}
Substituting $x=
3
$ in $P(x)=
2x^3+4x^2-10x-9
,$ then by the Remainder Theorem, the remainder is
\begin{align*}
P\left(3\right)&=
2\left(3\right)^3+4\left(3\right)^2-10\left(3\right)-9
\\&=
2\left(27\right)+4\left(9\right)-10\left(3\right)-9
\\&=
54+36-30-9
\\&=
51
.\end{align*}
Both solutions above show that the remainder, $P(a),$ is
\begin{align*}
P(a)=51
.\end{align*}
