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The given problem involves polynomial division. We need to divide:

\[
x^4 - 6x^3 + 8x^2 - 3x + 12 \quad \text{by} \quad x - 4
\]

and express the answer in the form:

\[
\text{Quotient} + \frac{\text{Remainder}}{x - 4}
\]

### Let's proceed with the division using synthetic division or long division.

#### Step-by-Step Long Division:
1. **First term division**: 
   - Divide \( x^4 \) by \( x \), which gives \( x^3 \).
   - Multiply \( x^3 \) by \( (x - 4) \), which gives \( x^4 - 4x^3 \).
   - Subtract \( (x^4 - 4x^3) \) from the original polynomial, resulting in \( -2x^3 + 8x^2 - 3x + 12 \).

2. **Second term division**:
   - Divide \( -2x^3 \) by \( x \), which gives \( -2x^2 \).
   - Multiply \( -2x^2 \) by \( (x - 4) \), which gives \( -2x^3 + 8x^2 \).
   - Subtract \( (-2x^3 + 8x^2) \) from the remaining polynomial, resulting in \( -3x + 12 \).

3. **Third term division**:
   - Divide \( -3x \) by \( x \), which gives \( -3 \).
   - Multiply \( -3 \) by \( (x - 4) \), which gives \( -3x + 12 \).
   - Subtract \( (-3x + 12) \) from the remaining polynomial, resulting in \( 0 \) (no remainder).

Thus, the division completes, and there is no remainder.

### Final Answer:
The quotient is:

\[
x^3 - 2x^2 - 3
\]

There is no remainder, so the final expression is:

\[
x^3 - 2x^2 - 3
\]

Let me know if you'd like more details or further clarification!

Here are 5 related questions you might consider:
1. How would the result change if the divisor was \( x + 4 \) instead of \( x - 4 \)?
2. Can you explain synthetic division and how it could be applied to this problem?
3. What are the differences between long division and synthetic division for polynomials?
4. How can we check if the result of the division is correct?
5. How does the degree of the polynomial affect the division process?

**Tip:** When dividing polynomials, it's helpful to arrange the terms in decreasing powers of \( x \), ensuring there are no missing terms.

Divide the polynomial x^4 - 6x^3 + 8x^2 - 3x + 12 by x - 4 and express the answer in the form: Quotient + Remainder/(x - 4).

Learn how to divide the polynomial x^4 - 6x^3 + 8x^2 - 3x + 12 by x - 4. This step-by-step solution explains the long division process and provides the final quotient and remainder. Suitable for Grades 10-12 students covering polynomial division in algebra.

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