Use long division to divide. Express any numbers as integers or simplified fractions. $$ \left(x^{5}+3 x^{4}+15 x^{2}-22 x+18\right) \div\left(x^{2}+2\right)=\square $$

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We wish to divide
$$
x^{5}+3 x^{4}+15 x^{2}-22 x+18
$$
by
$$
x^{2}+2.
$$
Since the dividend is missing the $$ x^3 $$ term, we rewrite it as
$$
x^5 + 3x^4 + 0x^3 + 15x^2 - 22x + 18.
$$

**Step 1.** Divide the leading term of the dividend by the leading term of the divisor:
$$
\frac{x^5}{x^2} = x^3.
$$
Multiply the entire divisor by $$ x^3 $$:
$$
x^3(x^2+2)=x^5+2x^3.
$$
Subtract this from the dividend:
$$
\begin{array}{rcl}
&&(x^5 + 3x^4 + 0x^3 + 15x^2 - 22x + 18)\[6mm]
&-& (x^5 + 0x^4 + 2x^3 + 0x^2 + 0x + 0)\[3mm]
\hline
&=& 3x^4 - 2x^3 + 15x^2 -22x +18.
\end{array}
$$

**Step 2.** Divide the new leading term $$ 3x^4 $$ by $$ x^2 $$:
$$
\frac{3x^4}{x^2}=3x^2.
$$
Multiply the divisor by $$ 3x^2 $$:
$$
3x^2(x^2+2)= 3x^4+6x^2.
$$
Subtract:
$$
\begin{array}{rcl}
&&(3x^4 - 2x^3 + 15x^2 -22x +18)\[3mm]
&-& (3x^4 + 0x^3 + 6x^2 + 0x + 0)\[3mm]
\hline
&=& -2x^3 + 9x^2 -22x +18.
\end{array}
$$

**Step 3.** Divide the leading term of the new remainder $$ -2x^3 $$ by $$ x^2 $$:
$$
\frac{-2x^3}{x^2}=-2x.
$$
Multiply the divisor by $$ -2x $$:
$$
-2x(x^2+2)= -2x^3 - 4x.
$$
Subtract:
$$
\begin{array}{rcl}
&&( -2x^3 + 9x^2 -22x +18)\[3mm]
&-& (-2x^3 + 0x^2 -4x + 0)\[3mm]
\hline
&=& 9x^2 -18x +18.
\end{array}
$$

**Step 4.** Divide the leading term $$ 9x^2 $$ by $$ x^2 $$:
$$
\frac{9x^2}{x^2}=9.
$$
Multiply the divisor by 9:
$$
9(x^2+2)= 9x^2+18.
$$
Subtract:
$$
\begin{array}{rcl}
&&(9x^2 -18x +18)\[3mm]
&-& (9x^2 +18)\[3mm]
\hline
&=& -18x.
\end{array}
$$
Since the degree of $$-18x$$ is less than the degree of the divisor, $$-18x$$ is the remainder.

**Final Answer:** The quotient is
$$
x^{3}+3x^{2}-2x+9,
$$
and the remainder is $$-18x$$. Thus, the division can be written as
$$
x^{3}+3x^{2}-2x+9-\frac{18x}{x^{2}+2}.
$$
The division of $$ x^{5}+3 x^{4}+15 x^{2}-22 x+18 $$ by $$ x^{2}+2 $$ results in a quotient of $$ x^{3}+3x^{2}-2x+9 $$ with a remainder of $$ -18x $$.

Use long division to divide. Express any numbers as integers or simplified fractions.

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