can you write the answer in the form of quotient +$$ \frac{\text{remainder}}{x-4} $$

Sure, let's rewrite the result in the required form:

Given:
\[ \frac{2x^3 - 8x^2 - 1}{x-4} \]

From the synthetic division, we obtained:
- Quotient: $2x^2 + 32$
- Remainder: $127$

Thus, the division can be expressed as:
\[ 2x^2 + 32 + \frac{127}{x-4} \]

So, the final answer in the form of quotient plus $\frac{\text{remainder}}{x-4}$ is:

\[ 2x^2 + 32 + \frac{127}{x-4} \]

Do you want more details or have any questions? Here are some related questions you might be interested in:

1. How is synthetic division different from long division in polynomials?
2. Can synthetic division be used for all types of polynomial division?
3. What are the advantages of using synthetic division?
4. How do you handle a remainder in polynomial division?
5. Can you apply synthetic division to polynomials with missing terms?
6. How do you interpret the results of synthetic division?
7. Can synthetic division be used for dividing by non-linear polynomials?
8. How does synthetic division relate to the Factor Theorem?

**Tip:** Always check for and include coefficients of any missing terms in the polynomial to avoid mistakes in synthetic division.

Explore how to divide polynomials using synthetic division to find the quotient and remainder. This problem involves the division of a polynomial by a linear polynomial (x-4), demonstrating advanced algebraic techniques. Suitable for advanced high school students and beyond.

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