College Algebra, Chapter 4, 4.3, Section 4.3, Problem 8

Two polynomials $P(x) = 2x^5 + 4x^4 - 4x^3 - x - 3$ and $D(x) = x^2 - 2$. Use either synthetic or long division to divide $P(x)$ by $D(x)$, and express $P$ in the form $P(x) = D(x) \cdot Q(x) + R(x)$.

Using Long Division







The process is complete at this point because $-x + 13$ is of lesser degree than the divisor $x^2 - 2$. We see that $Q(x) = 2x^3 + 4x^2 + 8$ and $R(x) = -x + 13$, So..


$
\begin{equation}
\begin{aligned}

2x^5 + 4x^4 - 4x^3 - x - 3 =& (x^2 - 2)(2x^3 + 4x^2 + 8) + (-x + 13)
\\
\\
2x^5 + 4x^4 - 4x^3 - x - 3 =& (x^2 - 2) (2x^3 + 4x^2 + 8) - x + 13


\end{aligned}
\end{equation}
$