Use long division to find the quotient $$ Q(x) $$ and the remainder $$ R(x) $$ when $$ P(x) $$ is divided by $$ d(x) $$ and express $$ P(x) $$ in the form $$ d(x) \cdot Q(x)+R(x) $$. $$ P(x)=x^{3}+1 x^{2}-27 x+232 $$ $$ P(x)=(x+8)(\square)+\square $$

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We want to divide
$$
P(x)=x^{3}+x^{2}-27x+232
$$
by
$$
d(x)=x+8.
$$

**Step 1.** Write down the coefficients of $$ P(x) $$: 
$$
1 \quad 1 \quad -27 \quad 232.
$$

**Step 2.** Since the divisor is $$ x+8 $$, we use $$ -8 $$ for synthetic division.

**Step 3.** Perform the synthetic division:

$$
\begin{array}{r|cccc}
-8 & 1 & 1 & -27 & 232 \
   &   & -8 & 56  & -232 \
\hline
   & 1 & -7 & 29  & 0 \
\end{array}
$$

- Bring down the $$ 1 $$.
- Multiply $$ 1 $$ by $$ -8 $$ to get $$ -8 $$, and add to $$ 1 $$ to get $$ -7 $$.
- Multiply $$ -7 $$ by $$ -8 $$ to get $$ 56 $$, and add to $$ -27 $$ to get $$ 29 $$.
- Multiply $$ 29 $$ by $$ -8 $$ to get $$ -232 $$, and add to $$ 232 $$ to get $$ 0 $$.

**Step 4.** From the synthetic division, the quotient is given by the coefficients $$ 1, -7, $$ and $$ 29 $$, so:
$$
Q(x)=x^{2}-7x+29.
$$

The remainder is $$ 0 $$, so:
$$
R(x)=0.
$$

**Step 5.** Write $$ P(x) $$ in the form
$$
P(x)=d(x)\cdot Q(x)+R(x).
$$

Thus:
$$
P(x)=(x+8)(x^{2}-7x+29)+0.
$$
$$ P(x) = (x + 8)(x^{2} - 7x + 29) $$

Use long division to find the quotient and the remainder when is divided by and express in the form .

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