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To solve this division of polynomials problem, we will use **polynomial long division**.

The division is:

\[
\frac{2x^3 - x^2 - 25x - 12}{x + 3}
\]

### Step-by-Step Solution:

1. **Divide the leading term of the numerator by the leading term of the denominator:**

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

   So, the first term in the quotient is \( 2x^2 \).

2. **Multiply \( 2x^2 \) by the entire denominator \( x + 3 \):**

   \[
   2x^2 \cdot (x + 3) = 2x^3 + 6x^2
   \]

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

   \[
   (2x^3 - x^2 - 25x - 12) - (2x^3 + 6x^2) = -7x^2 - 25x - 12
   \]

4. **Repeat the process with the new polynomial \( -7x^2 - 25x - 12 \):**

   \[
   \frac{-7x^2}{x} = -7x
   \]

   So, the next term in the quotient is \( -7x \).

5. **Multiply \( -7x \) by \( x + 3 \):**

   \[
   -7x \cdot (x + 3) = -7x^2 - 21x
   \]

6. **Subtract this from the current polynomial:**

   \[
   (-7x^2 - 25x - 12) - (-7x^2 - 21x) = -4x - 12
   \]

7. **Repeat the process with \( -4x - 12 \):**

   \[
   \frac{-4x}{x} = -4
   \]

   So, the next term in the quotient is \( -4 \).

8. **Multiply \( -4 \) by \( x + 3 \):**

   \[
   -4 \cdot (x + 3) = -4x - 12
   \]

9. **Subtract this from the current polynomial:**

   \[
   (-4x - 12) - (-4x - 12) = 0
   \]

Since the remainder is zero, the division is exact, and the quotient is:

\[
2x^2 - 7x - 4
\]

### Final Answer:

\[
\frac{2x^3 - x^2 - 25x - 12}{x + 3} = 2x^2 - 7x - 4
\]

Let me know if you'd like further clarification or details.

Here are 5 related questions:

1. What is synthetic division, and how does it differ from polynomial long division?
2. How do you verify the quotient in polynomial division?
3. What happens if there's a non-zero remainder after division?
4. How can polynomial division help in finding asymptotes of rational functions?
5. How would the solution change if the divisor was \( x - 3 \) instead of \( x + 3 \)?

**Tip:** When performing polynomial division, always ensure that all terms, including missing degrees of \( x \), are accounted for in both the dividend and the divisor.

Divide the polynomials: (2x^3 - x^2 - 25x - 12) / (x + 3)

Learn how to divide polynomials using polynomial long division with a detailed solution to the problem (2x^3 - x^2 - 25x - 12) divided by (x + 3). This guide covers the steps involved in polynomial division, from leading term division to handling remainders. Suitable for high school students in Grades 9-12.

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