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Let's solve this problem using synthetic division step by step.

### Problem
We are asked to divide the polynomial:
\[
2x^4 - 13x^3 - 13x^2 - 13x - 42
\]
by \(x + 2\), and express the result in the form:
\[
q(x) + \frac{r(x)}{d(x)}
\]
where \(q(x)\) is the quotient, \(r(x)\) is the remainder, and \(d(x)\) is the divisor.

### Synthetic Division Setup
In synthetic division, we need to use the *zero* of the divisor \(x + 2\), which is \(x = -2\).

We will divide the coefficients of the polynomial \(2x^4 - 13x^3 - 13x^2 - 13x - 42\). The coefficients are:
\[
[2, -13, -13, -13, -42]
\]

### Steps:

1. **Write the coefficients**: \(2, -13, -13, -13, -42\)
2. **Use \(x = -2\) as the synthetic divisor**.
3. Perform the synthetic division.

\[
\begin{array}{r|rrrrr}
-2 & 2 & -13 & -13 & -13 & -42 \\
   &    & -4  & 34   & -42  & 52  \\
\hline
   & 2  & -17 & 21   & -55  & 10  \\
\end{array}
\]

### Explanation:

- First, bring down the leading coefficient \(2\).
- Multiply \(2\) by \(-2\) and write it under \(-13\), giving \(-4\).
- Add \(-13 + (-4) = -17\).
- Multiply \(-17\) by \(-2\), giving \(34\), and write it under \(-13\), giving \(21\).
- Multiply \(21\) by \(-2\), giving \(-42\), and write it under \(-13\), giving \(-55\).
- Multiply \(-55\) by \(-2\), giving \(110\), and write it under \(-42\), giving a remainder of \(10\).

### Result:
The quotient is \(2x^3 - 17x^2 + 21x - 55\), and the remainder is \(10\).

Thus, the division can be written as:
\[
q(x) + \frac{r(x)}{d(x)} = (2x^3 - 17x^2 + 21x - 55) + \frac{10}{x+2}
\]

Would you like any additional clarification or steps explained in more detail?

---

### Related Questions:
1. How would synthetic division change if the divisor was \(x - 2\) instead of \(x + 2\)?
2. Can synthetic division be used for divisors of degree higher than 1?
3. What is the long division process for the same division problem?
4. How can you check if the quotient is correct using multiplication?
5. What is the role of the remainder in synthetic division?

**Tip:** In synthetic division, always remember that the divisor \(x + c\) becomes \(x = -c\) in the setup.

Use synthetic division to find the result when 2x^4 - 13x^3 - 13x^2 - 13x - 42 is divided by x + 2. Write your answer in the form q(x) + r(x)/d(x), where q(x) is the quotient, r(x) is the remainder, and d(x) is the divisor.

Learn how to use synthetic division to divide the polynomial 2x^4 - 13x^3 - 13x^2 - 13x - 42 by x + 2. This problem covers essential algebraic concepts and provides a step-by-step solution suitable for students in Grades 9-12.

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