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To solve the problem of dividing the polynomial \( 3x^4 - x^3 + 3x - 4 \) by \( x + 1 \), we need to perform polynomial division. The question asks for both the quotient and the remainder.

### Step 1: Set up the division
We are dividing \( 3x^4 - x^3 + 3x - 4 \) by \( x + 1 \).

### Step 2: Perform the division
1. **Divide the first term:** Divide \( 3x^4 \) by \( x \), which gives \( 3x^3 \).
2. **Multiply and subtract:** Multiply \( 3x^3 \) by \( x + 1 \), which gives \( 3x^4 + 3x^3 \). Subtract this from the original polynomial:
   \[
   (3x^4 - x^3 + 3x - 4) - (3x^4 + 3x^3) = -4x^3 + 3x - 4
   \]
3. **Repeat the process:**
   - Divide \( -4x^3 \) by \( x \), giving \( -4x^2 \).
   - Multiply \( -4x^2 \) by \( x + 1 \), resulting in \( -4x^3 - 4x^2 \).
   - Subtract this result:
     \[
     (-4x^3 + 3x - 4) - (-4x^3 - 4x^2) = 4x^2 + 3x - 4
     \]
   - Divide \( 4x^2 \) by \( x \), giving \( 4x \).
   - Multiply \( 4x \) by \( x + 1 \), resulting in \( 4x^2 + 4x \).
   - Subtract:
     \[
     (4x^2 + 3x - 4) - (4x^2 + 4x) = -x - 4
     \]
   - Divide \( -x \) by \( x \), giving \( -1 \).
   - Multiply \( -1 \) by \( x + 1 \), resulting in \( -x - 1 \).
   - Subtract:
     \[
     (-x - 4) - (-x - 1) = -3
     \]

### Step 3: Conclusion
The quotient is \( 3x^3 - 4x^2 + 4x - 1 \) and the remainder is \( -3 \).

### Final Answer
The correct option that matches this is:
\[ \boxed{\text{c. } 3x^3 - 4x^2 + 4x - 1; -3} \]

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**Related Questions:**
1. How do you perform polynomial division with more complex polynomials?
2. What is synthetic division and when is it applicable?
3. How do you find the remainder using the Remainder Theorem?
4. Can you explain how the Factor Theorem relates to polynomial division?
5. How does dividing polynomials differ from dividing numbers?

**Tip:** When performing polynomial division, always line up like terms carefully to avoid errors in subtraction and ensure the remainder is accurate.

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Learn how to divide the polynomial 3x^4 - x^3 + 3x - 4 by x + 1. This problem includes key concepts of polynomial division, quotient, and remainder calculations, suitable for students in Grades 10-12.

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