Use the long division method to find the result when 8, x, cubed, plus, 18, x, squared, plus, 19, x, plus, 68x 
3
 +18x 
2
 +19x+6 is divided by 2, x, plus, 12x+1

We are tasked with dividing the polynomial \( 8x^3 + 18x^2 + 19x + 6 \) by \( 2x + 1 \) using long division.

### Step 1: Divide the leading terms
We begin by dividing the leading term of the dividend \( 8x^3 \) by the leading term of the divisor \( 2x \):

\[
\frac{8x^3}{2x} = 4x^2
\]

Now, multiply the entire divisor \( 2x + 1 \) by \( 4x^2 \):

\[
(2x + 1) \times 4x^2 = 8x^3 + 4x^2
\]

### Step 2: Subtract the result from the original polynomial
Subtract \( 8x^3 + 4x^2 \) from the original polynomial:

\[
(8x^3 + 18x^2 + 19x + 6) - (8x^3 + 4x^2) = 14x^2 + 19x + 6
\]

### Step 3: Divide the new leading term
Now divide the new leading term \( 14x^2 \) by \( 2x \):

\[
\frac{14x^2}{2x} = 7x
\]

Multiply the divisor \( 2x + 1 \) by \( 7x \):

\[
(2x + 1) \times 7x = 14x^2 + 7x
\]

### Step 4: Subtract again
Subtract \( 14x^2 + 7x \) from \( 14x^2 + 19x + 6 \):

\[
(14x^2 + 19x + 6) - (14x^2 + 7x) = 12x + 6
\]

### Step 5: Divide the next term
Now divide \( 12x \) by \( 2x \):

\[
\frac{12x}{2x} = 6
\]

Multiply the divisor \( 2x + 1 \) by \( 6 \):

\[
(2x + 1) \times 6 = 12x + 6
\]

### Step 6: Subtract again
Subtract \( 12x + 6 \) from \( 12x + 6 \):

\[
(12x + 6) - (12x + 6) = 0
\]

### Final Answer:
The quotient is \( 4x^2 + 7x + 6 \) and the remainder is 0.

Thus, the result of the division is:

\[
\boxed{4x^2 + 7x + 6}
\]

Would you like more details, or do you have any questions?

Here are five related questions:
1. How do you handle division when the remainder is non-zero?
2. Can you apply synthetic division instead of long division for this problem?
3. What happens when dividing polynomials with missing terms?
4. How would the process change if the divisor had a higher degree?
5. How can you verify the correctness of the division?

**Tip:** When performing long division of polynomials, always align terms by their degree to avoid mistakes in subtraction.

Use the long division method to find the result when 8x^3 + 18x^2 + 19x + 6 is divided by 2x + 1.

Learn how to solve the polynomial division of 8x^3 + 18x^2 + 19x + 6 divided by 2x + 1 using long division. Follow a detailed step-by-step guide, perfect for high school students studying algebra. Suitable for Grades 10-12.

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