Let's perform the division of the polynomial \((2x^3 + x^2 - 13x + 6)\) by \((x + 3)\) using polynomial long division.

### Step-by-Step Solution:

1. **Divide the leading term** of the dividend by the leading term of the divisor:
   \[
   \frac{2x^3}{x} = 2x^2
   \]
   This will be the first term of the quotient.

2. **Multiply** \(2x^2\) by \((x + 3)\):
   \[
   2x^2(x + 3) = 2x^3 + 6x^2
   \]
   Now, subtract \((2x^3 + 6x^2)\) from the original polynomial:
   \[
   (2x^3 + x^2 - 13x + 6) - (2x^3 + 6x^2) = -5x^2 - 13x + 6
   \]

3. **Divide the leading term** of the new polynomial by the leading term of the divisor:
   \[
   \frac{-5x^2}{x} = -5x
   \]
   This is the next term of the quotient.

4. **Multiply** \(-5x\) by \((x + 3)\):
   \[
   -5x(x + 3) = -5x^2 - 15x
   \]
   Subtract \((-5x^2 - 15x)\) from the current polynomial:
   \[
   (-5x^2 - 13x + 6) - (-5x^2 - 15x) = 2x + 6
   \]

5. **Divide the leading term** of the new polynomial by the leading term of the divisor:
   \[
   \frac{2x}{x} = 2
   \]
   This is the final term of the quotient.

6. **Multiply** \(2\) by \((x + 3)\):
   \[
   2(x + 3) = 2x + 6
   \]
   Subtract \((2x + 6)\) from the current polynomial:
   \[
   (2x + 6) - (2x + 6) = 0
   \]

### Final Answer:
The quotient is:
\[
2x^2 - 5x + 2
\]

Let me know if you'd like more details or explanations.

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Here are 5 related questions to explore:
1. How do you perform synthetic division for polynomials?
2. Can you divide a polynomial by a binomial of higher degree, and how?
3. What is the Remainder Theorem, and how does it apply to polynomial division?
4. How do you factor a cubic polynomial?
5. What are the uses of polynomial division in calculus?

**Tip:** Polynomial division can be checked by multiplying the quotient by the divisor and comparing it to the original dividend.

Explore how to divide the polynomial (2x³ + x² - 13x + 6) by (x + 3) using polynomial long division. This problem breaks down algebraic concepts such as cubic polynomials and polynomial division. Suitable for high school students in Grades 9-12.

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