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$$  $$step by step solution answer correctly

Let's go through each problem step by step, performing polynomial long division and finding the quotient, remainder, zeros, and roots.

### 1. $ \frac{x^3 - 8x^2 - x + 12}{x - 2} $

**Step 1:** Divide the first term of the dividend by the first term of the divisor:

\[
\frac{x^3}{x} = x^2
\]

**Step 2:** Multiply the entire divisor by this quotient:

\[
x^2 \cdot (x - 2) = x^3 - 2x^2
\]

**Step 3:** Subtract this from the original polynomial:

\[
(x^3 - 8x^2 - x + 12) - (x^3 - 2x^2) = -6x^2 - x + 12
\]

**Step 4:** Repeat the process with the new polynomial:

\[
\frac{-6x^2}{x} = -6x
\]
\[
-6x \cdot (x - 2) = -6x^2 + 12x
\]
\[
(-6x^2 - x + 12) - (-6x^2 + 12x) = -13x + 12
\]

**Step 5:** Repeat again:

\[
\frac{-13x}{x} = -13
\]
\[
-13 \cdot (x - 2) = -13x + 26
\]
\[
(-13x + 12) - (-13x + 26) = -14
\]

**Result:**

- Quotient: $ x^2 - 6x - 13 $
- Remainder: $ -14 $

### 2. $ \frac{x^2 + 1}{x - 1} $

**Step 1:** Divide the first term:

\[
\frac{x^2}{x} = x
\]

**Step 2:** Multiply:

\[
x \cdot (x - 1) = x^2 - x
\]

**Step 3:** Subtract:

\[
(x^2 + 1) - (x^2 - x) = x + 1
\]

**Step 4:** Continue:

\[
\frac{x}{x} = 1
\]
\[
1 \cdot (x - 1) = x - 1
\]
\[
(x + 1) - (x - 1) = 2
\]

**Result:**

- Quotient: $ x + 1 $
- Remainder: $ 2 $

### 3. $ \frac{x^4 - 3x^3 + x}{x^2 + 1} $

**Step 1:** Divide:

\[
\frac{x^4}{x^2} = x^2
\]

**Step 2:** Multiply:

\[
x^2 \cdot (x^2 + 1) = x^4 + x^2
\]

**Step 3:** Subtract:

\[
(x^4 - 3x^3 + x) - (x^4 + x^2) = -3x^3 - x^2 + x
\]

**Step 4:** Continue:

\[
\frac{-3x^3}{x^2} = -3x
\]
\[
-3x \cdot (x^2 + 1) = -3x^3 - 3x
\]
\[
(-3x^3 - x^2 + x) - (-3x^3 - 3x) = -x^2 + 4x
\]

**Step 5:** Continue:

\[
\frac{-x^2}{x^2} = -1
\]
\[
-1 \cdot (x^2 + 1) = -x^2 - 1
\]
\[
(-x^2 + 4x) - (-x^2 - 1) = 4x + 1
\]

**Result:**

- Quotient: $ x^2 - 3x - 1 $
- Remainder: $ 4x + 1 $

### 4. $ \frac{x^3 + 1}{x - 1} $

**Step 1:** Divide:

\[
\frac{x^3}{x} = x^2
\]

**Step 2:** Multiply:

\[
x^2 \cdot (x - 1) = x^3 - x^2
\]

**Step 3:** Subtract:

\[
(x^3 + 1) - (x^3 - x^2) = x^2 + 1
\]

**Step 4:** Continue:

\[
\frac{x^2}{x} = x
\]
\[
x \cdot (x - 1) = x^2 - x
\]
\[
(x^2 + 1) - (x^2 - x) = x + 1
\]

**Step 5:** Continue:

\[
\frac{x}{x} = 1
\]
\[
1 \cdot (x - 1) = x - 1
\]
\[
(x + 1) - (x - 1) = 2
\]

**Result:**

- Quotient: $ x^2 + x + 1 $
- Remainder: $ 2 $

---

Would you like further explanation on any particular step, or have any questions on how to interpret these results?

### Follow-up Questions:
1. How does the remainder affect the overall division when solving polynomials?
2. What is the significance of the quotient in polynomial division?
3. How can synthetic division be used as an alternative to long division?
4. What happens if the divisor is a higher degree polynomial than the dividend?
5. Can you apply these methods to divide polynomials with coefficients that are not integers?

### Tip:
Always double-check each step in polynomial division, especially subtraction, to ensure accuracy in your quotient and remainder calculations.

Explore detailed step-by-step solutions for polynomial long division problems including finding quotient, remainder, and roots. Suitable for advanced high school students and those preparing for college-level mathematics.

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