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Let's solve this problem using synthetic division.

We are tasked with dividing the polynomial \( x^4 - 8x^3 + 12x^2 + 12x - 38 \) by \( x + 4 \).

### Step 1: Set up synthetic division

To begin, we will use synthetic division. The divisor is \( x + 4 \), so we will divide by \( -4 \) (the opposite of the coefficient of \( x \) in the divisor).

We take the coefficients of the polynomial \( x^4 - 8x^3 + 12x^2 + 12x - 38 \) which are:
\[ 1, -8, 12, 12, -38 \]

### Step 2: Perform synthetic division

Now, perform the synthetic division step by step:

1. Bring down the leading coefficient \( 1 \).
2. Multiply by \( -4 \) and add to the next coefficient:

\[
\begin{array}{r|rrrrr}
-4 & 1 & -8 & 12 & 12 & -38 \\
   &   & -4 & 48 & -240 & 912 \\
\hline
   & 1 & -12 & 60 & -228 & 874 \\
\end{array}
\]

The quotient is given by the terms from the bottom row, excluding the remainder. So, the quotient is:
\[ q(x) = x^3 - 12x^2 + 60x - 228 \]

And the remainder is \( r(x) = 874 \).

### Step 3: Express the result

We now express the result in the form:

\[
q(x) + \frac{r(x)}{d(x)} = x^3 - 12x^2 + 60x - 228 + \frac{874}{x + 4}
\]

### Final Answer:
\[
x^3 - 12x^2 + 60x - 228 + \frac{874}{x + 4}
\]

Would you like further details on any part of the process or a specific explanation? Here are 5 related questions to consider:

1. What are the steps involved in performing synthetic division for higher degree polynomials?
2. Why do we use \( -4 \) instead of \( 4 \) when dividing by \( x + 4 \)?
3. Can synthetic division be applied when dividing by polynomials of degree higher than 1?
4. How does synthetic division compare with long division in terms of efficiency?
5. How can you verify the result of synthetic division?

**Tip:** Synthetic division works best when dividing by binomials of the form \( x - c \), but for higher degree divisors, polynomial long division is typically used.

Use synthetic division to find the result when x^4 - 8x^3 + 12x^2 + 12x - 38 is divided by x + 4.

Learn how to use synthetic division to solve the polynomial division of x^4 - 8x^3 + 12x^2 + 12x - 38 by x + 4. Follow step-by-step instructions, with detailed explanations of key algebraic concepts. Suitable for students in Grades 10-12.

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