Use the long division method to find the result when 4, x, cubed, plus, 12, x, squared, plus, 19, x, plus, 74x 
3
 +12x 
2
 +19x+7 is divided by 2, x, plus, 12x+1.

To divide the polynomial \( 4x^3 + 12x^2 + 19x + 7 \) by \( 2x + 1 \) using the long division method, we follow these steps:

### Step 1: Divide the first term
We begin by dividing the leading term of the dividend \( 4x^3 \) by the leading term of the divisor \( 2x \):

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

This gives us the first term of the quotient.

### Step 2: Multiply the divisor by the first term of the quotient
Now, multiply the entire divisor \( 2x + 1 \) by \( 2x^2 \):

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

### Step 3: Subtract the result from the original polynomial
Next, subtract \( 4x^3 + 2x^2 \) from the original polynomial \( 4x^3 + 12x^2 + 19x + 7 \):

\[
(4x^3 + 12x^2 + 19x + 7) - (4x^3 + 2x^2) = 10x^2 + 19x + 7
\]

### Step 4: Divide the next term
Now, divide the leading term of the new polynomial \( 10x^2 \) by the leading term of the divisor \( 2x \):

\[
\frac{10x^2}{2x} = 5x
\]

This gives the next term of the quotient.

### Step 5: Multiply the divisor by the new term
Now, multiply \( 5x \) by \( 2x + 1 \):

\[
5x \cdot (2x + 1) = 10x^2 + 5x
\]

### Step 6: Subtract the result from the current polynomial
Subtract \( 10x^2 + 5x \) from \( 10x^2 + 19x + 7 \):

\[
(10x^2 + 19x + 7) - (10x^2 + 5x) = 14x + 7
\]

### Step 7: Divide the next term
Now, divide \( 14x \) by \( 2x \):

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

This gives the next term of the quotient.

### Step 8: Multiply the divisor by the new term
Multiply \( 7 \) by \( 2x + 1 \):

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

### Step 9: Subtract the result from the current polynomial
Subtract \( 14x + 7 \) from \( 14x + 7 \):

\[
(14x + 7) - (14x + 7) = 0
\]

Thus, the remainder is 0, and the division is complete.

### Final Answer:
The quotient is:

\[
2x^2 + 5x + 7
\]

Would you like further details or explanations on any of these steps?

Here are 5 related questions you could explore:
1. How is polynomial long division similar to numerical long division?
2. What happens if there is a remainder in polynomial division?
3. Can synthetic division be used to divide by non-linear divisors?
4. How do you perform synthetic division for polynomials?
5. What is the significance of the quotient and remainder in polynomial division?

**Tip:** Always align terms by their degree (powers of \(x\)) when performing polynomial long division for a clearer process.

Use the long division method to find the result when 4x^3 + 12x^2 + 19x + 7 is divided by 2x + 1.

Learn how to divide the polynomial 4x^3 + 12x^2 + 19x + 7 by 2x + 1 using the long division method. This detailed step-by-step guide covers key algebraic concepts and is perfect for students in Grades 10-12.

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