Example problems demonstrating polynomial division using the division algorithm.

Problem: Division of Polynomials Using the Division Algorithm

The task involves performing polynomial long division for two problems. Let's solve each one step by step.

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#### Problem 1:
Divide \( 5x^2 + 3x - 2 \) by \( x + 1 \).

Step 1: Set up the division.
\[
\begin{array}{r|rr}
x + 1 & 5x^2 + 3x - 2 \\
\end{array}
\]

Step 2: Divide the leading term of the dividend by the leading term of the divisor.
- Leading term of the dividend: \( 5x^2 \)
- Leading term of the divisor: \( x \)
\[
\frac{5x^2}{x} = 5x
\]
So, the first term of the quotient is \( 5x \).

Step 3: Multiply the entire divisor by this term and subtract from the dividend.
\[
5x \cdot (x + 1) = 5x^2 + 5x
\]
Subtract:
\[
(5x^2 + 3x - 2) - (5x^2 + 5x) = 5x^2 + 3x - 2 - 5x^2 - 5x = -2x - 2
\]

Step 4: Bring down the next term.
The next term is already there: \(-2\).

Step 5: Repeat the process.
- Divide the leading term of the new dividend (\(-2x\)) by the leading term of the divisor (\(x\)):
\[
\frac{-2x}{x} = -2
\]
So, the next term of the quotient is \(-2\).

Step 6: Multiply the entire divisor by this term and subtract.
\[
-2 \cdot (x + 1) = -2x - 2
\]
Subtract:
\[
(-2x - 2) - (-2x - 2) = -2x - 2 + 2x + 2 = 0
\]

Final Quotient and Remainder:
The quotient is \( 5x - 2 \) and the remainder is \( 0 \).

---

#### Problem 2:
Divide \( 6x^3 + 11x^2 - 31x + 15 \) by \( 3x - 2 \).

Step 1: Set up the division.
\[
\begin{array}{r|rrrr}
3x - 2 & 6x^3 + 11x^2 - 31x + 15 \\
\end{array}
\]

Step 2: Divide the leading term of the dividend by the leading term of the divisor.
- Leading term of the dividend: \( 6x^3 \)
- Leading term of the divisor: \( 3x \)
\[
\frac{6x^3}{3x} = 2x^2
\]
So, the first term of the quotient is \( 2x^2 \).

Step 3: Multiply the entire divisor by this term and subtract from the dividend.
\[
2x^2 \cdot (3x - 2) = 6x^3 - 4x^2
\]
Subtract:
\[
(6x^3 + 11x^2 - 31x + 15) - (6x^3 - 4x^2) = 6x^3 + 11x^2 - 31x + 15 - 6x^3 + 4x^2 = 15x^2 - 31x + 15
\]

Step 4: Bring down the next term.
The next term is already there: \(-31x\).

Step 5: Repeat the process.
- Divide the leading term of the new dividend (\(15x^2\)) by the leading term of the divisor (\(3x\)):
\[
\frac{15x^2}{3x} = 5x
\]
So, the next term of the quotient is \(5x\).

Step 6: Multiply the entire divisor by this term and subtract.
\[
5x \cdot (3x - 2) = 15x^2 - 10x
\]
Subtract:
\[
(15x^2 - 31x + 15) - (15x^2 - 10x) = 15x^2 - 31x + 15 - 15x^2 + 10x = -21x + 15
\]

Step 7: Bring down the next term.
The next term is already there: \(+15\).

Step 8: Repeat the process.
- Divide the leading term of the new dividend (\(-21x\)) by the leading term of the divisor (\(3x\)):
\[
\frac{-21x}{3x} = -7
\]
So, the next term of the quotient is \(-7\).

Step 9: Multiply the entire divisor by this term and subtract.
\[
-7 \cdot (3x - 2) = -21x + 14
\]
Subtract:
\[
(-21x + 15) - (-21x + 14) = -21x + 15 + 21x - 14 = 1
\]

Final Quotient and Remainder:
The quotient is \( 2x^2 + 5x - 7 \) and the remainder is \( 1 \).

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Final Answers:

1. For \( 5x^2 + 3x - 2 \div (x + 1) \):
\[
\boxed{5x - 2}
\]

2. For \( 6x^3 + 11x^2 - 31x + 15 \div (3x - 2) \):
\[
\boxed{2x^2 + 5x - 7 \text{ with remainder } 1}
\]