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To solve the division of the polynomial \( \frac{x^3 + x}{x - 1} \), we can perform polynomial long division. Let's go step-by-step:

### Step 1: Set up the division.
We are dividing \( x^3 + x \) by \( x - 1 \). First, write it in division form:
\[
\frac{x^3 + 0x^2 + x + 0}{x - 1}
\]

### Step 2: Divide the first term.
Divide the leading term of the numerator by the leading term of the denominator:
\[
\frac{x^3}{x} = x^2
\]
So, the first term in the quotient is \( x^2 \).

### Step 3: Multiply and subtract.
Now multiply \( x^2 \) by \( x - 1 \):
\[
x^2(x - 1) = x^3 - x^2
\]
Subtract this from the original polynomial:
\[
(x^3 + 0x^2 + x + 0) - (x^3 - x^2) = x^2 + x
\]

### Step 4: Divide the next term.
Divide the leading term of the new polynomial by the leading term of the denominator:
\[
\frac{x^2}{x} = x
\]
So, the next term in the quotient is \( x \).

### Step 5: Multiply and subtract again.
Now multiply \( x \) by \( x - 1 \):
\[
x(x - 1) = x^2 - x
\]
Subtract this from the current polynomial:
\[
(x^2 + x) - (x^2 - x) = 2x
\]

### Step 6: Divide the next term.
Divide the leading term of the new polynomial by the leading term of the denominator:
\[
\frac{2x}{x} = 2
\]
So, the next term in the quotient is \( 2 \).

### Step 7: Multiply and subtract one more time.
Now multiply \( 2 \) by \( x - 1 \):
\[
2(x - 1) = 2x - 2
\]
Subtract this from the current polynomial:
\[
(2x + 0) - (2x - 2) = 2
\]

### Step 8: Write the final answer.
The quotient is \( x^2 + x + 2 \) and the remainder is \( 2 \), so the division result is:
\[
x^2 + x + 2 + \frac{2}{x - 1}
\]

The correct choice is **d.** \( x^2 + x + 2 + \frac{2}{x - 1} \).

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Do you need further details or have any questions about this? 

Here are 5 related questions to expand your understanding:

1. What is the difference between polynomial long division and synthetic division?
2. How do you perform synthetic division on \( \frac{x^3 + x}{x - 1} \)?
3. How would the division change if the denominator were \( x + 1 \) instead of \( x - 1 \)?
4. What happens when the remainder is 0 in polynomial division?
5. Can long division be used to divide polynomials with higher-degree terms in the denominator?

**Tip:** Always arrange the terms of both the dividend and divisor in descending powers of \( x \) to ensure smooth division steps.

Divide polynomials using long division or synthetic division. Write \(\frac{x^3 + x}{x - 1}\) in algorithm form.

Learn how to divide the polynomial \(x^3 + x\) by \(x - 1\) using long division or synthetic division. This detailed solution guides you through each step of the process, making it ideal for students in Grades 9-12.

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