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We need to use polynomial long division to divide \( 8x^3 + 26x^2 + 25x + 6 \) by \( 2x + 3 \).

Let's perform the steps:

1. **Divide the leading term** of \( 8x^3 \) by the leading term of \( 2x \): 
   \[
   \frac{8x^3}{2x} = 4x^2
   \]
   This is the first term of the quotient.

2. **Multiply** \( 4x^2 \) by \( 2x + 3 \):
   \[
   4x^2 \cdot (2x + 3) = 8x^3 + 12x^2
   \]
   Now, subtract \( 8x^3 + 12x^2 \) from \( 8x^3 + 26x^2 \):
   \[
   (8x^3 + 26x^2) - (8x^3 + 12x^2) = 14x^2
   \]

3. **Divide the leading term** of \( 14x^2 \) by the leading term of \( 2x \):
   \[
   \frac{14x^2}{2x} = 7x
   \]
   This is the second term of the quotient.

4. **Multiply** \( 7x \) by \( 2x + 3 \):
   \[
   7x \cdot (2x + 3) = 14x^2 + 21x
   \]
   Now, subtract \( 14x^2 + 21x \) from \( 14x^2 + 25x \):
   \[
   (14x^2 + 25x) - (14x^2 + 21x) = 4x
   \]

5. **Divide the leading term** of \( 4x \) by the leading term of \( 2x \):
   \[
   \frac{4x}{2x} = 2
   \]
   This is the third term of the quotient.

6. **Multiply** \( 2 \) by \( 2x + 3 \):
   \[
   2 \cdot (2x + 3) = 4x + 6
   \]
   Now, subtract \( 4x + 6 \) from \( 4x + 6 \):
   \[
   (4x + 6) - (4x + 6) = 0
   \]

Thus, the quotient is:
\[
4x^2 + 7x + 2
\]

There is no remainder, so the result of the division is \( 4x^2 + 7x + 2 \).

Would you like further clarification or details on any step?

Here are some related questions:
1. What happens if the divisor has a degree higher than 1?
2. How do you perform synthetic division in comparison to long division?
3. Can you divide polynomials with remainders using long division?
4. What are some applications of polynomial long division in calculus?
5. How can you check the result of a polynomial division?

**Tip**: You can always verify your division by multiplying the quotient by the divisor and ensuring it equals the original polynomial.

Use the long division method to find the result when 8x^3 + 26x^2 + 25x + 6 is divided by 2x + 3.

Explore a step-by-step solution to the polynomial division of 8x^3 + 26x^2 + 25x + 6 by 2x + 3 using long division. This tutorial covers key algebraic techniques and polynomial division theorems, ideal for high school students in Grades 10-12.

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