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The given problem requires synthetic division of the polynomial \(\frac{x^4 - 16}{x + 2}\).

### Step-by-Step Solution:

1. **Rewrite the Division Expression**: We are dividing \(x^4 - 16\) by \(x + 2\). Synthetic division is typically used when dividing by a linear divisor of the form \(x - c\). So, for \(x + 2\), we take \(c = -2\).

2. **Set up the synthetic division**: 
   - The polynomial \(x^4 - 16\) is missing terms for \(x^3\), \(x^2\), and \(x\), so we need to represent it as: 
   \[
   x^4 + 0x^3 + 0x^2 + 0x - 16
   \]
   - Write the coefficients: \( [1, 0, 0, 0, -16] \).
   - Use \(c = -2\) as the divisor.

3. **Synthetic Division Process**:
   - Bring down the first coefficient \(1\).
   - Multiply \(1\) by \(-2\), giving \(-2\). Add this to the next coefficient, \(0\), yielding \(-2\).
   - Multiply \(-2\) by \(-2\), giving \(4\). Add this to the next coefficient, \(0\), yielding \(4\).
   - Multiply \(4\) by \(-2\), giving \(-8\). Add this to the next coefficient, \(0\), yielding \(-8\).
   - Multiply \(-8\) by \(-2\), giving \(16\). Add this to the last coefficient, \(-16\), yielding \(0\).

4. **Result**:
   The quotient is \(x^3 - 2x^2 + 4x - 8\), and the remainder is \(0\), meaning the division is exact.

Thus, the result of the synthetic division is:
\[
x^3 - 2x^2 + 4x - 8
\]

Let me know if you need further clarification or additional details.

### Follow-up Questions:
1. What is the general process for synthetic division?
2. How does synthetic division differ when the divisor is \(x - c\) vs \(x + c\)?
3. Why do we need to fill in missing terms with zeros when performing synthetic division?
4. How can we check if the result of synthetic division is correct?
5. What other methods, besides synthetic division, can be used for polynomial division?

### Tip:
Always ensure to include all terms, even those with a coefficient of zero, when setting up synthetic division to avoid mistakes.

Use synthetic division to divide the polynomials (x^4 - 16)/(x + 2).

Learn how to use synthetic division to divide the polynomial (x^4 - 16) by (x + 2). This detailed step-by-step solution explains the process, key algebraic concepts, and synthetic division for higher-degree polynomials. Suitable for students in Grades 10-12.

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